^(Please note I've moved my account to /u/Bealoideas, so if you have any further questions on this topic, I would be happy to answer messages sent there.)

I've only studied poets and bardic poetry tangentially as it related to my own area, but I can give you some basic detail. The word for poet itself, file, comes from a proto-Celtic word for `to see', so file originally were sort of like seers. File and Druí were probably interchangeable at one point, so Druids and Poets might have been one continuum... though that's a whole other topic of discussion. File were an acceptable pre-Christian class in Christian Ireland, there is a story of St. Patrick blessing the mouth of a poet so he can compose better religious verse. The file as a class were very highly ranked in a society that was very strict about hierarchy, and in fact they remained so up until the 17th Century. Part of this was due to education... I remember reading that during the medieval period, 21 years was thought to be the requisite length of study for one to become a file.

Part as well due to the power of satire in Irish society. Irish society in the medieval and pre-Christian eras was very concerned with outward appearance, and very little with the interior self. The most striking example I can think of is Nuada, legendary king of the Tuatha Dé Danann, who during the Cath Maige Tuired has his hand dismembered in combat, and is rendered immediately and incontestably ineligible to rule, not due to disability, but because his body was no longer perfect. This is still reflected to an extent in the Irish language, where qualities and statuses are most typically 'on' or 'at' a person, rather than within them. Regardless there was a general fear of being satirized in early Ireland, and the fear only grew with greater position. In the Exile of the Sons of Uisliu for example, a story that is a heartbreaking encapsulation of the raw-deal that women had in Iron Age society, a female character Leborcham, tutors the ill-fated Dierdre, against the wishes of her captor Conchobar, but he is powerless to stop her because of her skill as a satirist.

The position then was a high one in society, Fergus Kelly's A Guide to Early Irish Law would have more detail on this, but I believe the rank was similar to a bishop or chieftain. Kelly's book also talks about satire, some styles of which were stated to be magical in nature - and this is in a law tract keep in mind.

In terms of what form the poetry took, that is getting out of my wheelhouse somewhat. Bardic poetry was a very, very conservative tradition, so the language being written was different from the spoken Irish language of the time. As a counter to the aforementioned satire, praise-poetry was a common genre, with poets on retainer to Kings and chiefs of various kinds, to glorify them in all kinds of ways. In structure it was...very complex. Irish is a very dense poetic language, and the technical requirements for poetry were high.

There was plenty of poetry that would be closer to the modern concept, that is poems about things strictly for their own sake... I was just recently reading this poem in fact, a sixteenth Century lament by Laoisioch Mac an Bhaird for the cutting down of a thorn tree (tree-cutting can be very taboo in Irish tradition, even today).

On the Cutting Down of an Ancient Tree

Hail to thee, O hill yonder: at thy fall I am not joyous; thy brown thorn is a cause of woe, the smooth stem that was wont to be seen above thee.

The thorn of acclamation, a torment to all, I used to see as a place of assembly: the cutting of the branch, my day of sorrow! the state of the land is baser thereafter.

My heart in my breast is sad for thy ancient tree, 0 hill yonder; the stem from which I was wont to see each tract, thy smooth thorn I see not there.

And so one for several verses. That translation I think is Fergus Kelly from his book on Bardic poetry. Ironically the (probably) most famous early Irish poem, was not written by a poet at all, but by a Christian monk writing in the margins of a book of miscellany, about his cat, Pangur Bán. That's getting off the topic though: There is certainly a lot of Bardic poetry, and a lot of material has been written about it. Osborn Bergin and Fergus Kelly are the scholars I would start with if you'd like to read more about it. I'm sorry I can't get into more detail on the poetry itself, I'd be veering out of my depth.

Airgialla has posted a pretty good answer concerning who the file actually were, so I'll address this part of your question:

One thing I specifically wondered is how the dismantling of the 'class' system of Gaelic Ireland affected these poets, since they occupied an important position in society, what happened to them after? Did the poetic tradition still survive and have an impact on later traditions such as aislings?

The file depended on the patronage of the Irish aristocracy for their livelihoods, and following the rise of the dynastic kingship system in the 4th-5th centuries AD served to legitimate that aristocracy's tenuous rule. Irish poets and historians crafted intricate genealogies which legitimized the rule of alien warrior-aristocracies over subject people, or created fictional links between the supposed ancestors of subject dynasties and those of the heads of large dynastic kingships/confederacies. Donnchadh O Corrain has called these poets and historians the 'mandarin caste' of early medieval Ireland, who upheld the prevailing social and political institutions through indirect means - which is hardly surprising because the medieval Irish were wont to do everything through indirect means.

Fast forward a few centuries to the Cromwellian conquest of Ireland in the mid-17th century, when the indigenous Irish aristocracy was more or less extirpated in Ireland. The file still produced propaganda for the aristocracy before the conquest, but it seems to have served a different purpose as dynastic kingship, as it was practiced in the early medieval period, had sort of died out. By this period it seems to have served the purpose of legitimating the Irish aristocracy's rule by differentiating them from the lowly and obscene Irish peasantry who the bardic caste held in contempt. Poetry from this time was written to a very formulaic and classical model which completely disappeared following the Cromwellian conquest. This is because the Irish aristocracy who were accustomed to the classical style of Irish poetry had almost instantly disappeared, depriving the file of their hereditary patronage, and forcing them to pursue their livelihoods amongst the peasantry who they hated, and who weren't accustomed to classical Irish verse. Interestingly, one distant modern-day descendant of the bardic class still exists; the seanchaí or seanachie. These are itinerant or village storytellers who memorize lots of stories orally, and who still exist in some rural parts of Ireland.