What is meant by setting the "wheel" of Dharma in motion?

I have something interesting to share. By the grace of Buddha I guess I have found the full meaning of chariot symbolism (including the wheel). This quote from Jāṇussoṇibrāhmaṇa Sutta (SN 45.4) illustrates the meaning of the chariot and wheel symbol:

This is what the Blessed One said. Having said this, the Fortunate One, the Teacher, further said this:

“Its qualities of faith and wisdom
Are always yoked evenly together.
Shame is its pole, mind its yoke-tie,
Mindfulness the watchful charioteer.

“The chariot’s ornament is virtue,
Its axle jhana, energy its wheels;
Equanimity keeps the burden balanced,
Desirelessness serves as upholstery.

“Good will, harmlessness, and seclusion:
These are the chariot’s weaponry,
Forbearance its armour and shield,
As it rolls towards security from bondage.

“This divine vehicle unsurpassed
Originates from within oneself.
The wise depart from the world in it,
Inevitably winning the victory.”

And on which path the wise departs from the world in it? On the noble eight fold path. It all fits properly.

What are the three phases and twelve aspects of Dhamma?

Copying extracts from this text and putting them into a numbered list:

1st truth:

1. This is the dukkha ariyasacca': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
2. 'Now, this dukkha ariyasacca is to be completely known': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
3. 'Now, this dukkha ariyasacca has been completely known': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.

2nd truth:

4. 'This is the dukkha·samudaya ariyasacca': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
5. 'Now, this dukkha·samudaya ariyasacca is to be abandoned': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
6. 'Now, this dukkha·samudaya ariyasacca has been abandoned': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.

3rd truth:

7. 'This is the dukkha·nirodha ariyasacca': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
8. 'Now, this dukkha·nirodha ariyasacca is to be personally experienced': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
9. 'Now, this dukkha·nirodha ariyasacca has been personally experienced': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.

4th truth:

10. 'This is the dukkha·nirodha·gāminī paṭipadā ariyasacca': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
11. 'Now, this dukkha·nirodha·gāminī paṭipadā ariyasacca is to be developed': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.
12. 'Now, this dukkha·nirodha·gāminī paṭipadā ariyasacca has been developed': in me, bhikkhus, in regard to things unheard before, the eye arose, the ñāṇa arose, the paññā arose, the vijjā arose, the light arose.

So 12 aspects, consisting of 3 phases for each of the 4 truths.

Why the noble Truth of the origin of suffering is to be abandoned ? (I think craving needs to be abandoned and not the Truth)

Yes. Instead of:

This noble truth of the origin of suffering is to be abandoned

... it might be better to translate as, something like, "This (truth which is the) origin of suffering is to be abandoned" ... i.e. "craving", as you said: it's craving that's to be abandoned.

The Pali in question is,

‘Idaṃ dukkha·nirodhaṃ ariyasacca’

Piya Tan writes:

Norman adds that

what the Buddha said was that pain should be known, its origin given up, its cessation realized, and the path to its cessation practised. Woodward therefore did not go far enough. He should have suggested the removal of the word ariya-saccaṁ from all four items in the ‘gerundival’ set.”
(Norman 1982:385)

There appears to be a copy of that paper here ... which analyses the grammar and compares it to other suttas in which the Truths are described.

Reading that paper might inspire several insights: e.g. that it's difficult to translate Pali accurately; that people have put a lot of work into trying to translate them; that maybe we should have some humility in asserting which is the 'right' translation, and/or "know the truths for ourselves" and not depend only on scholars.

I think this paper concludes that "Noble truth" is maybe a later addition to the sutta: that the phrase "dukkha·samudaya" (origin of suffering) later became known as "dukkha·samudaya ariyasacca" (noble truth of origin of suffering), and that the new phrase became used even in places like the one you mentioned where it isn't appropriate (not appropriate because, as you said, it's not truth that's to be abandoned).

What is meant by setting the "wheel" of Dharma in motion? Why is it called a wheel?

Here is my interpretation based on "things I heard here and there" (so not official):

In this ancient metaphor, a human society with its traditional ways is compared with a chariot ("yana", a word that also means "marching ahead"). Presumably, some sage or a visionary sets the chariot in motion, and then it rolls on for some time until it stops. A wheel-turning monarch is a leader who then supports and maintains this traditional order of things.

Apparently the number of wheels in a chariot varies over time. In more ancient texts they are presumably listed as "artha" (success or prosperity) and "dharma" (in this context referring to morality or virtue) balancing each other as society moves from one generation to next. Newer texts are said to have added a third wheel of "kama" (enjoyment), followed by a fourth wheel of "moksha" (the spiritual quest for liberation). It is unclear how many "wheels" were recognized in Buddha's time, 2 or 3, but presumably the wheel of "moksha" or liberation was not added until after the Buddha's lifetime.

We do not know if Buddha knew about this metaphor when he spoke about turning the wheel, but if he did, he would be referring to one specific wheel out of 2 or 3 - the wheel of Dharma.

By turning the wheel of Dharma he was renewing or re-setting in motion the tradition of virtue and high ethics, that has seriously degraded by his time (judging by his criticism of brahmins with their corrupted morality). This tradition was then to be followed by the subsequent generations, propelling society to a greater virtue and thus enabling the growth of prosperity. (Which is kinda what happened if we look at the spread of Buddhism throughout the ancient Asian countries)

The cyclic action here refers to the subsequent generations of people adopting the tradition and passing it on. Each generation of Buddhist students and teachers basically repeats Buddha's path: getting inspired by the possibility of Enlightenment, working on it in one's real life, achieving the realization, and passing some guidelines and insights to the next generation of students.