What did the Buddha do, and what does a teacher do, to teach?

Pali Buddhism is not non-labeling. Pali Buddhism is the giving up of craving, attachment & egoism.

Pali Buddhism forbids monks displaying psychic powers to laypeople (Vinaya Rule 8 & Kevatta Sutta).

The Guru is not so important in Pali Buddhism.

Occasionally, the Pali suttas report the Buddha using his psychic powers to help a monk who had an obstacle (examples: Capala Sutta & Sona Sutta) but, in general, the Pali suttas report monks received a teaching and then went in solitude and practised alone.

In the Buddha's final days, it was reported he said: "The Dhamma Teaching will be the guru".

Now the Blessed One spoke to the Venerable Ananda, saying: "It may be, Ananda, that to some among you the thought will come: 'Ended is the word of the Master; we have a Master no longer.' But it should not, Ananda, be so considered. For that which I have proclaimed and made known as the Dhamma and the Discipline, that shall be your Master when I am gone.

Maha-parinibbana Sutta

Mayahana (Tibetan) Buddhism, which is influenced by Hinduism, is different to Pali Buddhism.

like a realized guru, by his mere will can lever us in higher jhanic state

Instead of that, the examples (of the Buddha's psychic power) told in suttas tend to be rather of the "divine eye" and "divine ear" (seeing what's happening elsewhere, and knowing the mind-state of another person) and the ability to travel instantly.

Using a psychic power to help someone (as narrated in the canon) tends to consist of telling the person what they need to hear, telling hem something that's to their benefit -- i.e., essentially, a dhamma-talk of one kind or another.

Perhaps it's not surprising that suttas consist mostly of dhamma-talks (because suttas tend to be a collection of the "thus have I heard" stories).

Even so, using "will to lever us in higher jhanic state" isn't (for example here) numbered among the psychic powers.

When buddha gave diksha, what he did: is that was the imparting of knowledge?

I guess that giving diksha might be equivalent to, in Buddhism, either a "refuge" ceremony (for lay people) or "ordination" (for monks and nuns).

I don't think that the ceremony itself is intended/expected to impart knowledge: and that people are supposed to understand in advance (before the ceremony) what path they're starting on.

how the teacher is useful?

I think that (as well as teaching dhamma and meditation) teachers sometimes they offer explanations about the discipline, and advice about whatever day-to-day problem might arise; for example:

A Western monk at WatBa Pong became frustrated by the difficulties of practice and the detailed and seemingly arbitrary rules of conduct the monks had to follow. He began to criticize other monks for sloppy practice and to doubt the wisdom of Achaan Chah's teaching. At one point, he went to Achaan Chah and complained, noting that even Achaan Chah himself was inconsistent and seemed often to contradict him self in an unenlightened way.

Achaan Chah just laughed and pointed out how much the monk was suffering by trying to judge others around him. Then he explained that his way of teaching is very simple: "It is as though I see people walking down a road I know well. To them the way may be unclear. I look up and see someone about to fall into a ditch on the right-hand side of the road, so I call out to him, 'Go left, go left' Similarly, if I see another person about to fall into a ditch on the left, I call out, 'Go right, go right!' That is the extent of my teaching. Whatever extreme you get caught in, whatever you get attached to, I say, 'Let go of that too.' Let go on the left, let go on the right. Come back to the center, and you will arrive at the true Dharma."

I think they're also meant to be a good (i.e. reliable and perhaps inspiring, "admirable") example of how to practice.

-- When the teachings of Buddha are crystal clear i.e 1+1=2, then no need to be labelled.

You are right, but people have different levels of prajna (deep intuitive understanding of how things work). Someone who is very mature, because of the work done earlier in this life or in previous lives, may not need any explanations, but someone who is just starting or very confused - may need lots of words and labeling. It depends on the student.

--When buddha gave diksha, what he did: is that was the imparting of knowledge?

Not all teaching is given in words. Some teaching is given in a more subtle form. The famous Flower Sermon is a legend of how Zen Buddhism started when Buddha gave what you might call "shaktipat" to his student Mahakashyapa.

--was that the imparting of knowledge?

It is a kind of information, yes - but not in the coarse form that we usually call "information". It is a subtle kind of information that we can also call "abstract energy". So yes, it is knowledge - but it is deep intuitive kind of knowledge, not the superficial knowledge conveyed in words.

--And in the path of Buddha, once known, is to be walked upon by individual: then, how the teacher is useful? Whether he pushes us ahead by using his jhanic powers, or resolves our difficulties?

It's a little bit of everything. Also, depends on the student and on the teacher of course. But usually it has elements of 1) behavioral training, 2) leading by example, 3) regular logical knowledge, 4) emotional inspiration, 5) deeper intuitive understanding, 6) dealing with real-life situations, 7) connecting the student with external sources of energy/information, 8) meditation.

--Because in Hinduism there is mention of shaktipat by guru. Is it the same in Buddhism also?

What you call "diksha" is called Refuge in Buddhism - and yes in that ceremony has an element of "shaktipat":

At that particular point, the energy, the power, and the blessing of basic sanity that has existed in the lineage for twenty-five hundred years, in an unbroken tradition and discipline from the time of Buddha, enters your system, and you finally become a full-fledged follower of buddhadharma. You are a living future buddha at that point.

In Tibetan Buddhism there is also "abhisheka" ceremony (empowerment), which is almost completely "shaktipat".

So yes, generally speaking there are some parallels, although there are differences too - the main difference being that in Buddhism we tend to be more "scientific" and much less respectful of what you would call Sacred Elements. In Buddhism we liberate ourselves using something like "mind science", so in this sense Buddhism is more modern.

Anyway, the role of the "teacher" is not as much "to teach" knowledge, as it is to teach the skills, mature the student's mind and deliver the student to the enlightened perspective. And yes, it involves transmitting something that cannot be explained in words. Although, you must understand that in Buddhism we do not have dualistic notion of "special energy" that is somehow different from normal stuff - we just say that this energy is a subtle/abstract aspect of regular things.

But yes, just like the law systems of different countries have similarities - because they are based on the actual human nature which is more or less the same everywhere (in terms of basic goodness, vices, desires, problems) -- similarly the different religions like Christianity, Hinduism, Buddhism etc. have many common elements and many overlaps - because they are reflections of True Dharma - the True Nature of Reality.

At least in the Theravada tradition, there is no mystical energy seed that needs to be transferred from the teacher to the disciple through a medium like touch or speech, in order to kick-start the disciple's spiritual growth.

Rather, the teacher teaches the Buddha's Dhamma (teachings) using his own knowledge, experience and insight, much like how a mathematics teacher teaches mathematics to others.

The teacher must have the following qualifications according to the Udayi Sutta:

"It's not easy to teach the Dhamma to others, Ananda. The Dhamma should be taught to others only when five qualities are established within the person teaching. Which five?

"(1) The Dhamma should be taught with the thought, 'I will speak step-by-step.'

"(2) The Dhamma should be taught with the thought, 'I will speak explaining the sequence [of cause & effect].'

"(3) The Dhamma should be taught with the thought, 'I will speak out of compassion.'

"(4) The Dhamma should be taught with the thought, 'I will speak not for the purpose of material reward.'

"(5) The Dhamma should be taught with the thought, 'I will speak without hurting myself or others.' (see Note)

"It's not easy to teach the Dhamma to others, Ananda. The Dhamma should be taught to others only when these five qualities are established within the person teaching."

Note:
According to the Commentary, "hurting oneself" means exalting oneself. "Hurting others" means putting other people down.