Partial answer only here; off the top of my head I don't know of any information regarding the social structure of the Elves in Aman (other than the fact that the Vanyar were considered above the others in some way), and I don't really have the time to dig through History of Middle-earth right now. But I can answer your other questions.
What role to the Valar play in Aman?
They are both teachers and rulers. From The Silmarillion we know that the Elves learned a great many things from the Valar and the Maiar; to give a few examples:
The Vanyar learn music from Manwë:
Manwë has no thought for his own honour, and is not jealous of his power, but rules all to peace. The Vanyar he loved best of all the Elves, and of him they received song and poetry; for poetry is the delight of Manwë, and the song of words is his music.
The Silmarillion III Quenta Silmarillion Chapter 1: "Of the Beginning of Days"
The Noldor learned smithcraft from Aulë:
Aulë it is who is named the Friend of the Noldor, for of him they learned much in after days, and they are the most skilled of the Elves; and in their own fashion, according to the gifts which Ilúvatar gave to them, they added much to his teaching, delighting to tongues and in scripts, and in the figures of broidery, of drawing, and of carving. The Noldor also it was who first achieved the making of gems; and the fairest of an gems were the Silmarils, and they are lost.
The Silmarillion III Quenta Silmarillion Chapter 1: "Of the Beginning of Days"
The Teleri learned sea-craft and (later) ship-building from Ossë the Maia:
Long [the Teleri] remained by the coasts of the western sea, and Ossë and Uinen came to them and befriended them; and Ossë instructed them, sitting upon a rock near to the margin of the land, and of him they learned all manner of sea-lore and sea-music. Thus it came to be that the Teleri, who were from the beginning lovers of water, and the fairest singers of all the Elves, were after enamoured of the seas, and their songs were filled with the sound of waves upon the shore.
[...]
Therefore Ulmo, submitting to the will of the Valar, sent to [the Teleri] Ossë, their friend, and he though grieving taught them the craft of ship-building; and when their ships were built he brought them as his parting gift many strong-winged swans. Then the swans drew the white ships of the Teleri over the windless sea; and thus at last and latest they came to Aman and the shores of Eldamar.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of Eldamar and the Princes of the Eldalië"
Celegorm, son of Fëanor, learned from Oromë the Hunter:
Celegorm went rather to the house of Oromë, and there he got great knowledge of birds and beasts, and all their tongues he knew. For all living things that are or have been in the Kingdom of Arda, save only the fell and evil creatures of Melkor, lived then in the land of Aman; and there also were many other creatures that have not been seen upon Middle-earth, and perhaps never now shall be, since the fashion of the world was changed.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of Eldamar and the Princes of the Eldalië"
However, the Valar (especially Manwë and Mandos) set the laws that the Eldar must follow, and arbitrate legal disputes. Some examples of this are noted in the question:
- Fëanor is banished from Tirion for threatening Fingolfin
- Manwë sets a ban on the Noldor returning to Aman
But there are other examples; the one that occurs to me immediately is on the matter of Finwë, Míriel, and Indis. A brief backstory on this dispute is given in The Silmarillion:
[I]n the bearing of her son Míriel was consumed in spirit and body; and after his birth she yearned for release from the labours of living. And when she had named him, she said to Finwë: 'Never again shall I bear child; for strength that would have nourished the life of many has gone forth into Fëanor.'
Then Finwë was grieved, for the Noldor were in me youth of their days, and he desired to bring forth many children into the Miss of Aman; and he said: 'Surely there is healing in Aman? Here all weariness can find rest.' But when Míriel languished still, Finwë sought the counsel of Manwë, and Manwë delivered her to the care of Irmo in Lórien. At their parting (for a little while as he thought) Finwë was sad, for it seemed an unhappy chance that the mother should depart and miss the beginning at least of the childhood days of her son.
‘It is indeed unhappy,’ said Míriel, 'and I would weep, if I were not so weary. But hold me blameless in this, and in all that may come after.’
She went then to the gardens of Lórien and lay down to sleep; but though she seemed to sleep, her spirit indeed departed from her body, and passed in silence to the halls of Mandos. The maidens of Estë tended the body of Míriel, and it remained unwithered; but she did not return.
The Silmarillion III Quenta Silmarillion Chapter 6: "Of Fëanor and the Unchaining of Melkor"
After his wife's death, Finwë is a bit upset because he wants to have more children, but an Elf is only allowed to have one spouse and the law took no account of a spouse who has died. What happened next is in Morgoth's Ring, and is a very long tale that I'm not going to reproduce here; I've discussed parts of it in my answer to Did Mandos make any decisions that didn't prove to be disastrous?.
What basically happened is that Mandos made the judgement that, if Míriel decided that she did not desire ever to return to her body, only then would the marriage be dissolved and Finwë could take another wife. The Valar debated this Statute, and ultimately found it to be just, and it passed into the laws of the Eldar.
I want to highlight one thing that Mandos says in this discussion, because it's highly relevant to the question:
'It is our part to rule Arda, and to counsel the Children, or to command them in things committed to our authority. Therefore it is our task to deal with Arda Marred, and to declare what is just within it.
History of Middle-earth X Morgoth's Ring Part 2: The Second Phase Chapter 3: "Laws and Customs of the Eldar" Of the Severance of Marriage
What I take away from this is that it's primarily the duty of the Valar to set the laws. This meshes with the evidence above of the Valar enforcing the law, because in each case they're adjudicating the first instance of a law needing to exist:
- Fëanor drawing his sword on Fingolfin is the first time the peace of Valinor is broken
- The ban on immigration is a response to the First Kinslaying1
- Míriel is the first Elf to die in Aman, so the Statute of Finwë and Míriel was a response to the first occurrence of needing such a statute
Does this then mean that the Kings of the Kindred apply these laws, once decided, to their own people? I don't know, and I don't think there's enough information to say.
1 Is this response maybe a bit over-broad? I would argue yes, since many of the Noldor who were blameless in the Kinslaying nonetheless fell under this ban. But we're not here to discuss whether a particular ruling is just or not
