Do Buddhists believe Buddha to be God? Do Buddhists practice idolatry?

This answer is from the perspective of Theravada Buddhism.

The Buddha is not The God or a god. He was a human being who sought to discover the solution to the suffering condition of sentient beings, and found it when he became enlightened at the age of 35. He taught the path to the end of suffering to people around him, who were willing to listen to him. He then passed away at the age of 80. His teachings were known as the Dhamma.

After passing away, the Buddha was not reborn in any realm. It is impossible to reach him through prayers or rituals.

Before he passed away, he told his disciple, Ananda (in DN16):

Therefore, Ananda, be islands unto yourselves, refuges unto yourselves, seeking no external refuge; with the Dhamma as your island, the Dhamma as your refuge, seeking no other refuge.

So, the Buddha did not tell his followers to pray to him for salvation. Rather he told them to depend on the Dhamma and their own effort for enlightenment.

The Buddha told Vakkali (in the Vakkali Sutta):

He who sees Dhamma, sees me; he who sees me sees Dhamma. Truly seeing Dhamma, one sees me; seeing me one sees Dhamma.

This means that after his passing, the Dhamma is the only "form" in which he remains present.

If that is the case, then why do people still have statues for the Buddha?

Again from DN16, the Buddha explained to Ananda, why it is beneficial to have a stupa for a Tathagata (a Buddha):

And why, Ananda, is a Tathagata, an Arahant, a Fully Enlightened One worthy of a stupa? Because, Ananda, at the thought: 'This is the stupa of that Blessed One, Arahant, Fully Enlightened One!' the hearts of many people will be calmed and made happy; and so calmed and with their minds established in faith therein, at the breaking up of the body, after death, they will be reborn in a realm of heavenly happiness. And so also at the thought: 'This is the stupa of that Paccekabuddha!' or 'This is the stupa of a disciple of that Tathagata, Arahant, Fully Enlightened One!' or 'This is the stupa of that righteous monarch who ruled according to Dhamma!' — the hearts of many people are calmed and made happy; and so calmed and with their minds established in faith therein, at the breaking up of the body, after death, they will be reborn in a realm of heavenly happiness. And it is because of this, Ananda, that these four persons are worthy of a stupa.

I would therefore extrapolate that the purpose of the statue of the Buddha is similar to the purpose of the stupa of a Buddha, which is, to be an inspiration for peace of mind and happiness.

Today, people in USA remember Martin Luther King, people in India remember Mahatma Gandhi and people in South Africa remember Nelson Mandela. They may have photos or statues of these revered persons put up in public places and treated in a respectable way. The statue of the Buddha is no different. They are there to serve as icons of inspiration.

The Buddha is no more supernaturally reachable through prayers and rituals, than King, Gandhi or Mandela.

The answer seems to be yes or no: depending on how you define "idolatry", and whether you consider the belief+intent or only the action+appearance. For example, Wikipedia says "yes",

According to Eric Reinders, icons and idolatry has been an integral part of Buddhism throughout its later history. Buddhists, from Korea to Vietnam, Thailand to Tibet, Central Asia to South Asia, have long produced temples and idols, altars and rosaries, relics to amulets, images to ritual implements. The images or relics of Buddha are found in all Buddhist traditions, but they also feature gods and goddesses such as those in Tibetan Buddhism.

Bhakti (called Bhatti in Pali) has been a common practice in Theravada Buddhism, where offerings and group prayers are made to Buddhist icons and particularly images of Buddha.

That Wikipedia section seems to be citing anglophone academics.

Conversely What Buddhists Believe, written by a monk, says no:

Although it is customary amongst Buddhists to keep Buddha images and to pay their respects to the Buddha, Buddhists are not idol worshippers. Idolatry generally means erecting images of unknown gods and goddesses in various shapes and sizes and to pray directly to these images. The prayers are a request to the gods for guidance and protection. The gods and goddesses are asked to bestow health, wealth, property and to provide for various needs; they are asked to forgive transgressions.

The 'worshipping' at the Buddha image is quite a different matter. Buddhists revere the image of the Buddha as a gesture to the greatest, wisest, most benevolent, compassionate and holy man who has ever lived in this world. It is a historical fact that this great man actually lived in this world and has done a great service to mankind. The worship of the Buddha really means paying homage, veneration and devotion to Him and what He represents, and not to the stone or metal figure.

The image is a visual aid that helps one to recall the Buddha in the mind [etc.]

Apparently the earliest phase of Buddhism was aniconic: see Aniconism in Buddhism.

I'd treat the above as an orthodox answer but it's possible that the answer (personal beliefs, people's hopes for answered prayer) may vary from person to person and/or from school to school.

In case you're interested, history includes a very similar question or concern, about whether other religions are idolatrous -- for example there is art and statues in many Christian churches; but they'll tell you they're not worshipping the statue itself, rather that they're praying to the saint (or Jesus) represented by the statue.

Even so there have been various "iconoclastic" movements or reforms at various times in history.

I can't tell you to what extent that's "like the Hindus" (you could ask that on Hinduism.SE).

Buddhists don't consider Buddha to be a god. Buddhists consider Buddha as a being beyond any god. That's why he is called "Satta Deva Manussanam" - the teacher to gods and humans.

The things you mentioned are forms of paying respect to the Buddha. If you learn the meanings of the stanzas Buddhists chant while worshipping, you will understand the purpose better.

The statues are used just as a representation of the Buddha. They know that those are not the real Buddha and that they are made of stone or wood. The concept is similar to people using framed pictures of their dead parents and lighting candles etc. as a form of respect and appreciation.

Old fashion hindu, brahma, was begun before buddhism, so people in buddha time usually worshiped to their god, before they change to be the buddha-follower.

Buddha just change them from god-believer/random-result/suffering-still-remain to cause&effect-professor/best-advantage/suffering-perfectly-vanishing.

So, if they normally worship god with incense, candles, flower, etc, buddha just change them to worship the cause&effect-professor, buddha&dhamma&saṅgha, instead.

Because idol and statue are what the follower try to imitate to do and to be the same. If they were still paying the worship to god, they will be god-believer, but if they change their habit to pay their worship to buddha/dhamma/saṅgha, they will be cause&effect-professor instead.

Example, can you change the homeless habit immediately? If you can't, how can you use their daily habit to change themselves?

This is the reason why the buddha is the best teacher, anuttaro purisadhammasārathi&bhagavā, because he can change the illogical homeless/illogical king/illogical people to be the perfect cause&effect-professors.

Another, Worshiping, offering, giving gift, donating, making statue, naming the road as the people name, and investing are just an exchange. We are acting like that because we expecting some effects.

So, the buddha just let them do the right exchange by leting them change their common action, god-worshiping, to the new right habit, cause&effect-worshiping, such as in Sutta. Ma. U. mahācattārīsakasutta, that the buddha taught the listener to worship to right reason, right relation of causes&effects, instead of god:

sāsavā puññabhāgiyā upadhivepakkā

• [saṅkhāra, kamma-bhava] right view depending on kilesa-vaṭṭa-paṭiccasamuppāda [kāma-āsava (kāma-upādāna), diṭṭhi-āsava (diṭṭhi-upādāna, sīlabata-upādāna), bhava-āsava (attavāda-upādāna), avijjā-āsava],
• right view that is kammavaṭṭa-paṭiccasamuppāda,
• right view that give vipākavaṭṭa-paṭiccasamuppāda.
  
  

1. atthi dinnaṃ [sukata­dukka­ṭā­naṃ kammānaṃ phalaṃ vipāko],

There is given-gift-action['s fruit and result of good and bad given-gift-actions].

2. atthi yiṭṭhaṃ [sukata­dukka­ṭā­naṃ kammānaṃ phalaṃ vipāko] ,

There is worshiped-action['s fruit and result of good and bad worshiped-actions].

3. atthi hutaṃ [sukata­dukka­ṭā­naṃ kammānaṃ phalaṃ vipāko],

There is given-oblation-action['s fruit and result of good and bad given-oblation-actions].

4. atthi sukata­dukka­ṭā­naṃ kammānaṃ phalaṃ vipāko,

There is fruit and result of good and bad actions.

5. atthi ayaṃ loko [sukata­dukka­ṭaṃ kammaṃ],

There is this-aggregate (=cause) [loka sutta].

6. atthi paro loko [phalaṃ vipāko],

There is another-aggregate (=effect).

7. atthi mātā [imā lokā sukata­dukka­ṭā kammāikā],

There are benefactor mother (=cause).

8. atthi pitā [imā lokā sukata­dukka­ṭā kammāikā],

There are benefactor father (=cause).

9. atthi [imā lokā sukata­dukka­ṭā] sattā opapātikā,

There are benefactor deva (such as died mother, died father).

10. atthi loke samaṇabrāhmaṇā sammaggatā sammāpaṭipannā ye imañca lokaṃ parañca lokaṃ sayaṃ abhiññā sacchikatvā pavedenti.

There are teachers who have done right path then have clearly viewed this-aggregate (=cause) and another-aggregate (=effect) by themselves.