In Mahayana Buddhism, do they expect everyone to become a Buddha? If so how can any Buddha help anyone to attain enlightenment?

Yes. In The Diamond Sutra:

...all living beings will eventually be led by me to the final Nirvana, the final ending of the cycle of birth and death...

This answer is just my speculation, based on experience of medicine not based on Buddhist references.

Isn't the whole point of becoming a Buddha to help others to attain enlightenment?

I guess I don't understand your analogy.

The way I think it works is,

• I'm ill and a doctor heals me
• I'm grateful for my health and grateful to the doctor
• I want to be able to better assure my own health, so I learn a bit of medicine
• I want others to be blessed with good health too (or I assume they want that); I assume that everyone will be happy when everyone is liberated from illness

Meanwhile, even the doctors encourage me to learn a bit of medicine too:

• Because they're busy and it's better to not require the doctors' attention unless you need them (i.e. better for everyone if yu're a bit healthy and independent yourself)
• Because prevention is better than cure (better to know enough medicine and hygiene to avoid getting sick, than to become sick and need to be healed by a doctor)
• Because the outcome (for the patient) is better when the patient (and their family) are relatively well-informed about medicine, and can understand and double-check and monitor the treatment.

In summary if I can help someone attain or maintain good health, isn't that a good thing?

There's also the idea that illnesses are a bit contagious. Helping the people around me to be healthy will help me, and helping myself helps the people around me (see also herd immunity).

It's like asking everyone in the country to become kings. Then there won't be any countrymen left to rule.

The Buddha decided not to be ruler.

Doctors too seem to me to be generally benevolent -- to the extent they want your obedience, it's your obedience to the dhamma (not to themselves) that they want, your understanding of and compliance with the prescriptions for well-being.

And a lot of countries aren't monarchies these days.

Then those future patients they are hoping to help would also have to refuse their help to try to become doctors themselves.

I think a big part of knowing medicine is knowing when someone needs to visit a doctor. Though I'm not a doctor I assume that even doctors don't refuse the help of other doctors when they need it.

I'm continuing this answer, from these comments that were originally posted under Andrei's answer:

CW -- Ordinary (i.e. non-Buddhist or pre-Buddhist) English language has a word "selfless", whose definition is "concerned more with the needs and wishes of others than with one's own; unselfish" with synonyms "unselfish, altruistic, self-sacrificing, self-denying; considerate, compassionate, kind, decent, noble, public-spirited; generous, magnanimous, ungrudging, unstinting, charitable, benevolent, liberal, open-handed, philanthropic". I guess that can conceal conceit (e.g. "I'm much more selfless than you are!") but IMO it is an obvious avenue, for people who hope to become or to remain unselfish.

SK -- @ChrisW "I'm much more selfless" is a contradiction in Buddhism. Because when you say "I am", there's already 'self' implied. To say that those who work towards Arahathship are selfish is a complete misunderstanding of the Dhamma and meditation.

I didn't intend to imply that Arahats are selfish, and sorry it sounded that way. What I meant was that even without or before Buddhism I might think it ordinary and obvious for there to be a yana based on helping others, especially once it's decided that "selflessness" (or anatta) is a good thing (a beneficial doctrine). I'm pretty sure "service" and "ministry" are also Christian ideals (Christians too sometimes puzzle, and sects' doctrines disagree with each other, over how man co-exists with the divine, but being selfless and aspiring to be like Jesus is I think a standard ideal, especially in the early Church).

I don't mean that Buddhism is Christianity, just that the yana may be (it seems to be) analagous or at least comparable, and so it doesn't sound like a too-absurd idea, though it is a lofty ideal. Perhaps I'm culturally (i.e. from birth) predisposed to that as an ideal (i.e. taught that it is an ideal), so it doesn't sound absurd nor difficult to accept as doctrine (though when it's examined in detail it may be found to be a cause for wide schisms in Christianity, e.g. here: "every Christian has equal potential to minister for God [...] stands in opposition to the concept of a spiritual aristocracy or hierarchy within Christianity")

See also Is there a compassion only Buddhist path?.

Finally I think what you're stuck (focused) on is the idea that there can only be one Buddha. But I guess the vow is less "me is going to become Buddha!", and is instead more like "sentient beings (without bounds) need salvation".

The Four Encompassing Vows

Masses [of] creatures, without-bounds,
[I/we] vow to save [them all].

Anxiety [and] hate, [delusive-desires] inexhaustible,
[I/we] vow to break [them all].

Dharma gates beyond-measure
[I/we] vow to learn [them all].

Buddha Way, unsurpassable,
[I/we] vow to accomplish [it]

SK -- Vipassana meditation is all about not seeing the world through ignorance and start taking each experience as it really is. In other words, stop doing wrong and start doing right. It has nothing to do with making a 'self' enlightened.

OK.

I think maybe this answer describes a beginning to the Mahayana.

For what it's worth I understand the vow (quoted above) to mean, not "making a self enlightened", but "acting in an unselfish (and unlimited) way" ... as well as "accomplishing" the path (which, "accomplishing the path", may be an ideal that's described in the suttas too).

It's worthy to think about in how far one is still just another Bodhisattva, no matter which wisdom or vehicle one calaims to be ones own... ;-)

It's always very tricky to like to stay in the same boot, isn't it?

And then a sublime smile on ones smart account might fade to getting disenchanted in regard of ones smart found, doesn't it? Well then, if not, take another turn, Boddhisattva to become one seeking maybe just after Awakening, a Bodhisatta!

Or should you be called a healed one, doctor for trade and livelihood?

Can one trace the classical "Bodhisattva(-ideal)" in the picture?

[Note: This is a gift of Dhamma, not meant for commercial use or other lower wordily gains by ways of exchange or trade.]

Terminology

To answer this question correctly it is important to understand that Prasangika-Madhyamika Mahayana distinguish the attainments of Sravakas, Pratyekabuddhas and Buddhas.

In short, Sravakas and Pratyekabuddhas are those beings who have attained nirvana with remainder in post-meditative equipoise. While in meditative equipoise it is said they have attained nirvana without remainder.

It is said that Buddhas on the other hand have attained nirvana without remainder regardless of whether they are in meditative equipoise.

So what is the difference between nirvana with and without remainder? Well, both include the complete cessation of true suffering and the elimination of the conception of true existence, but the nirvana with remainder is said to include obscurations to knowledge. In other words, Sravakas and Pratyekabuddhas have not obtained the omniscience of the Buddhas.

It is taught in Mahayana that only the nirvana of a Buddha (a non-abiding nirvana) eliminates the obscurations to knowledge that obscure omniscience.

To further distinguish, the Bodhisattva is said to embark on the path to full enlightenment (non-abiding nirvana) for the sake of suffering sentient beings. An Arya being is someone who it is said to have had a direct realization (non-conceptual) of emptiness while in meditative equipoise.

More detailed terminology can be found in this answer. Now, we are in a better position to answer your question(s).

In Mahayana Buddhism, do they expect everyone to become a Buddha?

Yes, although I think expect is not the right word. It is more like everyone has the (eventual) capacity to achieve non-abiding nirvana or to become a completely perfected fully awakened being. However, in this very life it is more appropriate for some beings to strive for something less than the Bodhisattva ideal.

If so, how can any Buddha help anyone attain enlightenment? Isn't the whole point of becoming a Buddha to help others to attain enlightenment?

The Buddha is in the perfect position to help others attain enlightenment, but it is still up to each sentient being to do the work. In other words, the Buddha cannot do the work for us. That said, Buddhas are unparalleled in their ability to help sentient beings. Their abilities far exceed those of Sravakas and Pratyekabuddhas in that they possess omniscience and therefore know perfectly how to help transmigrating beings.

It's like asking everyone in the country to become kings. Then there won't be any countrymen left to rule.

Buddhas are not kings though nor do they rule over sentient beings. Neither are Sravakas or Pratyekabuddhas for that matter.

To give another analogy, imagine patients refusing to follow doctor advice saying that they want to become doctors first to help cure others.

Ah, I think this gets to the crux of the question! If I understand you correctly, the question is how can Bodhisattvas refuse to become Sravakas or Pratyekabuddhas and instead insist upon becoming Buddhas instead? If everyone followed the Mahayana, then we'd all be refusing to become enlightened before we helped everyone else to do so first?? Like if we were all too naively polite to pass anyone else through the door of liberation, then no one at all would pass??!!

I think this is a misunderstanding. The Bodhisattva path does not foreclose Bodhisattvas from reaching liberation as an Sravaka or Pratyekabuddha before continuing on to become a fully enlightened perfect being. That'd be absurd and silly. We also do not posit that after one becomes an Sravaka or Pratyekabuddha in this very life that with the breakup of the body that these beings cease to exist in any way and are therefore unable to continue on to attain full and complete enlightenment. Much the opposite! The difference is the motivation. The Bodhisattva wished to attain enlightenment and pacification of the afflictions in this very life, but not for oneself; the motivation is to do so for the benefit of other sentient beings.

The whole point of the Bodhisattva path is to cultivate the wish to help sentient beings in the best way possible and then to realize that becoming a Buddha is the perfect way to accomplish this. Moreover, it is only through the Bodhisattva motivation that is possible for one to attain the full and completely perfected enlightenment.

Then those future patients they are hoping to help would also have to refuse their help to try to become doctors themselves. At the end of the day, no patient makes use of any doctor and no doctor cures any patient. So why become a doctor in the first place? :)

I hope you agree that the above explanation renders this a misunderstanding. It is not the case that Bodhisattva's play an absurd game of "no, no, after you! no, no, after you!" with each other on the path to a completely perfected full enlightenment :)

The Bodhisattva Vows include a vow not to disparage The Fundamental Vehicle (Theravada)

Here is Venerable Thubten Chodron on the importance of the Fundamental Vehicle even for those aspiring towards full enlightenment:

In a previous vow, we want to abandon criticizing the Mahayana and throwing it out. Here, it is criticizing the Theravada and saying, “Oh, we don’t need to practice any of those teachings. We’re great Mahayana practitioners! The Theravada tradition doesn’t help you abandon attachment. It won’t get you liberated. We don’t have to practice those teachings.” That’s completely incorrect. The Mahayana is built upon the foundation of the Theravada. Everything that you find in the Theravada is found in the Mahayana. It is like the building blocks. Some people think that if you practice this tradition, you don’t practice the other one. It is not like this.

If you practice Mahayana, you have to practice what’s taught in the Theravada vehicle.

Emphasis mine.

Also, if you look at this answer you'll see that there is a specific vow that Bodhisattvas take which abandons "believing and saying that followers of the Mahayana should remain in cyclic existence and not try to attain liberation from afflictions."

^{Title question asked by OP on Jan 14 at 11:42 ~}

In Mahayana Buddhism, do they expect everyone to become a Buddha? If so how can any Buddha help anyone to attain enlightenment?

Yes. For the 1st question.

I do not understand in logic, how the following 2nd question had to be asked. Explanation - by replacing terminologies with daily words:

In Medical Science (Mahayana Buddhism), do they expect everyone to become a healthy person (Buddha)? If so how can any Medical Scientist (Buddha) help anyone to gain (attain) health (enlightenment)?

In this replacing terminology practice - purposefully making it easy for OP used doctor-patient analogy, cannot one see the burble of asking the 2nd question? Or, it's still too hard to grasp once the healthy person replaced by medical scientist? Are scientists not persons, or a person not scientist? ...Maybe need more learning on Logic and Sets then.

^{Question content elaborated by OP on Jan 14 at 11:42 ~}

In Mahayana Buddhism, do they expect everyone to become a Buddha? If so, how can any Buddha help anyone attain enlightenment? Isn't the whole point of becoming a Buddha to help others to attain enlightenment?

This the same as title question, intriguing to read; yet to utilize what asked, enlightenment for human beings is a gradual process. 2500 years after the Buddha entered Nirvana, there are 7.442 billion (2016), only maybe a fraction as small as paramāṇu (micro-dust) enlightened, plenty works to do. The OP seems worried there not enough unenlightened persons left to let the Buddha enlighten them :).

It's like asking everyone in the country to become kings. Then there won't be any countrymen left to rule.

This is not a legitimate analogy. Countrymen vs Kings, ruler vs ruled... this is a Samsaric formula, which Buddhism, the pinnacle of Buddhism, dedicated to transform. For the Buddhist ideal, there no kings, neither ruled; everyone is his own king, his own master; not mastered by Mara, or other, or society, or greed/anger/delusion, or any mental affliction... etc.

To give another analogy, imagine patients refusing to follow doctor advice saying that they want to become doctors first to help cure others. Then those future patients they are hoping to help would also have to refuse their help to try to become doctors themselves. At the end of the day, no patient makes use of any doctor and no doctor cures any patient. So why become a doctor in the first place?:)

Again it's amusing to instance patient refusing doctor advice by claiming himself doctor to cure other first (??? / I commented, but being moved to chat). A patient aspired to be doctor doesn't make him a patient refused doctor advice, unless his mentally afflicted; a doctor can be sick so he a patient, a patient can be cured and then learning medication became the doctor... etc. etc.

The heart of Mahayana Buddhism is to aspire to become the Buddha, for that the realization greatest potentiality of any sentient being, especially Human Being. To be a Buddha is to regain the totality of one's beingness without defilement (but of course there more profound teachings about defilement is not defiled for what originally perfect can never be scathed). As a Buddha, naturally he will aspire for the greatest benefit of all sentient beings, the greatest most wondrous existence is being a Buddha - so a Buddha will help all sentient beings to attain Buddha-hood. Buddha = compassion = selfless love, just generalization; the mind of the Buddha is beyond comprehension, it said.

To produce a Buddha, the "everyone" (unenlightened beings) is the ingredient. Through helping others to attain liberation, a Buddha is born. Said in 《普賢行願品》 (The Aspiration of Bodhisattva Samantabhadra) that this whole is like the Bodhi tree, the leaves and branches the sentient beings, the root Prajna-paramita, the fruit the Buddha. Without any one part, that's not a tree any more. Help others attaining liberation from simply ridding illness (many real Buddhist practitioners possessed this healing power naturally), be their trustworthy friends, providing services, giving virtuous advice... to enlighten them, all go gradually or spontaneously.

Like, if you real wealthy, that must be when you can give to anyone in need; you a kind person, that must be when you showing your kindness to others; you good doctor, when have cured a patient with terminal illness. Likewise a Buddha a Buddha is by helping sentient attaining enlightenment.

There will always sentient beings be born into Samsara, for Samsara denoted cyclical recurring. Unless all sentient awake all at once in this Great Dream of Maya. Obviously from the beginning-less and endless of spatial/temporal dimension, this never happened, else we won't exist (are we the dream dreaming our existences or we dreaming the dream?? - Zhuang Zhou and his butterfly metaphor :). For the fact this never happened, I can surely infer it will never happen too, simply, Samsaric time is beginning-less and endless (I know many won't be able to understand at here - I will be misunderstood, again!!). So those who don't understand Mahayana will intercept here laden their lethal attack - liberation of all sentient beings a futile aspiration!!

Yes, futile; but that almost ensured there will be plenty of unenlightened sentient to be liberated (solved the OP's intrigue ;). It ensures anyone aspired to be the Buddha will have plenty and more than enough ingredient to work on :). Further, a futile attempt doesn't diminish the one who held the greatest aspiration - in fact the glorious of such spirit. Isn't there a popular gig: Impossible = I'm possible :)?

It could go on and on... to cut short and conclude, the enigmatic Mahayanists proclaim,

眾生無邊誓願度；(innumerable sentient I vow to ferry across)

煩惱無盡誓願斷；(endless fetters I vow to cut off)

法門無量誓願學；(countless Dharma gates I vow to study)

佛道無上誓願成。(unsurpassable Buddha Path I vow to accomplish)

These original Chinese Mahayanists are likely, crazy. Yeah.

Do not confuse the Chinese Mahayana Bodhisattva Samantabhadra with Tibetan Samantabhadra. Chinese one the handsome young man with beautiful clothes and adornments riding on white elephant; whilst Tibetan is blue-skin naked in copulating with his mate.

Chinese Mahayana Samantabhadra

Tibetan Samantabhadra 

In Mahayana Buddhism, they assume that everyone is already a Buddha. Enlightenment (bodhi) is self-existing, it is our natural state. It is our true nature, our primordial face. The only problem is, we got alienated from it, lost connection with it - but only at the superficial level. Deep inside we are always enlightened, all we need is to connect the layers and remember.

So when someone attains enlightenment, they see everyone as enlightened already. They also see the confusion, or the illusion of confusion. And that's what they help the others to overcome.

It's like, you wanted to become free from slavery and you thought the only way to do that is to become a king. But then you realize that this sort of thinking is still a slavery thinking. You realize that in fact we're all already free, that slavery was a lie that we believed in only because of ignorance. But as you go around on your white horse helping others get this, some may mistake you for a king. And then it's up to you to either tell them the truth, or to play a king for their benefit, like an adult playing Santa for the sake of kids.