I've never heard of what happened to the culture after the fall of Rome and the western half.
I know afterwards a eastern half emperor retook Rome by sending troops thru the aqueducts and the inhabitants of Rome were happy but none of the books or articles I've read about Rome has ever really discussed the culture going away.
Is Italian considered Roman? I wouldn't think it is it seems very different.
It didn't GO anywhere. There was no mass genocide of the Roman people living in the former empire. Their modes of life of course changed to cope with different circumstances of course but the people did not all of a sudden wake up and decide to be Visigothic rather than Roman. If you wished you could even make a reasonably compelling argument that moving into Roman lands doomed the barbarian tribes. They voraciously adopted Roman customs, methods of government, language and religion and tried their best to identify themselves with the former empire as a source of legitimacy. I am certainly not arguing that Clovis or Odoacer were Romans, far from it. Just that the assumption that "Roman culture" suddenly popped like a soap bubble when the empire vanished is a bit over simple. Cultural change, like the transition from Roman to Italian or Roman to French, was a process that took hundreds of years.
Hi! I answered a similar question a couple of months ago about the fate of aristocratic culture in the Gallic provinces. If you are interested in changes in culture in the post-Roman world, I would recommend reading:
I should say I am not much use on late ancient Italy, but I am happy to answer any follow up questions that you may have. The simple answer is that Roman civic cultures and institutions did not disappear, but became increasingly regional in scope and focus as local aristocrats worked together with new arrivals (the 'barbarians') to forge new communities of power and patronage in the post-imperial West.