The best source for this particular area of Post-Roman European society, is the works of an actual Roman, an Italian political writer by the name of Flavius Magnus Aurelius Cassiodorus Senator, or Cassiodorus for short (Strangely, 'Senator' was actually part of his name). His letters give a huge amount of practical information about how Roman society functioned under the rule of the formerly 'barbarian' Goths. Found online here:

http://www.gutenberg.org/files/18590/18590-h/18590-h.htm#Page_141

He was a statesman during the reign of the Ostrogothic king of Italy, Theodoric the Great, and was an important advisor to the Ostrogothic court all throughout Theodoric's reign and into the reign of his successors, including the reconquest by Emperor Justinian of the Eastern Roman Empire. As a Roman in Ostrogothic Italy, he wrote extensively about the laws and works of Theodoric, who he venerated heavily, and this includes the important issue of what was to happen to the Roman population the Ostrogoths now ruled.

Theodoric, probably under advice from his Roman advisors, but also as a result of his own childhood as a hostage in Constantinople, treated the Roman population relatively fairly and allowed them their own courts, markets, semi-independent town councils and other elements of the Roman state. The army, however, was almost exclusively to be made up of Ostrogothic warriors. Theodoric, as seen in Cassiodorus' works, proclaimed that Romans and Goths were to be 'separated' and that, whilst the Ostrogoths would defend Italy and rule justly under their king, the Romans would be 'free' to practice art, music, economic matters and generally keep the Roman society of Italy alive.

Regarding the lack of 'Roman' naming in Italy, that came about as a result of a long period of indeterminate length by which formerly Roman peoples would take on the cultural practices, dress, names and general attitudes of their 'barbarian' rulers. This happened all across the former Empire, from Britannia under the Angles and Saxons, to Hispania under the Visigoths and Frankish Gaul. This process took years, and some texts I've read claim that 'Roman' society carried on uninterrupted in some parts all the way to the 7th century AD, when the Islamic conquests, various plagues and a general breakdown in long-distance economic links made a Roman lifestyle untenable. One source I read at university, which unfortunately I can't remember the name of, claimed that, in Italy 'Poor Romans would dress as Goths, and rich Goths as Romans', so essentially, whilst the general Roman people began taking on their new masters' ways, their 'barbarian' kings would take on Roman ways.

Sources:

The Roman Empire Divided: 400-700: The Post-Roman World, 400-700, by John Moorhead

The Restoration of Rome: Barbarian Popes & Imperial Pretenders, by Peter Heather

THE LETTERS OF CASSIODORUS BEING A CONDENSED TRANSLATION OF THE VARIAE EPISTOLAE OF MAGNUS AURELIUS CASSIODORUS SENATOR trans. Thomas Hodgkin