The clear division between the Classical concept of Rome and the feeling of considering oneself "Roman" comes around the VIII-IX centuries, when the Lombards have already settled most of Italy for two hundred years already and king Charlemagne conquered their kingdom, "of the Lombards".
After it, the XII century is a moment where political life and struggles against external influences, for example the German emperors claiming Northern Italy as part of the HRE, did help the idea that "Italy was inhabited by Italians" so to speak, adding to the fact that linguistically most of Italy's dialects take shape during these years.
The 1200s would be the very first time the idea of being Italian takes proper form since we start having works of literature and documents written in vernacular languages and a strong increase in political factionalism and the creation of a stereotype of average Italian. Most important cities in Europe have at least one road named "Lombard Street" or something like that, since Italian merchants and bankers (most of whom came from the city-states of Central and Northern Italy) exported their financial structures in other regions of the contintent.

For the rest of your questions, the city of Rome in 476 CE and later would be much less populated, less well maintained, less wealthy and less important than its eastern step brother, Constantinople, but still a huge and well defended city to the point that centuries of Lombard rule couldn't attempt to conquer it. Still in the XII and XIII centuries Rome would boast some of the Western world's most efficient fortifications (the Aurelian walls). The Pope and the remnants of the wealthies senatorial families would control and govern the city, often wrestling the title of pope among themselves. The movie Pope Joan shows this in a rather nice manner.

As for the language, Latin would begin its metamorphosis in the current regional dialects around the turn of the millennium. Before the late 1100s we don't really know much about he sermo vulgaris, the commoner's language (the very earliest written document featuring vulgar language is a series of court acts from 960 to 963 about a litigation between a monastery and a count). The current branch of Italian that we recognize as such and is taught in schools and used as official language, is the dialect of Florence used in the literary works of three poets and writers from that city in the 1300s: Dante Alighieri, Giovanni Boccaccio and Francesco Petrarca.

For the culture, in mainland Italy people kept wearing what they would have worn in 476 CE well into the kingdom of Charlemagne, although they would partially drop the Roman Law as the Germanic Law of the ruling Lombards would be used alongside it and gradually replace it until the early XI century. There are clear evidences of two bodies of laws being applied on an ethnic base, to the point that king Rothari in 643 codified in a written form the oral tradition of his people.
This occurred in the center and north of the peninsula. In some areas of the south, Byzantine rule would last until the Norman conquest of 1085 and influence much in the way of laws and political structure (well into the 1200s the royal official in the government of Naples would be called stratigotus in the Greek manner).

As for the rest of Roman provinces during the Middle Ages, they would follow a similar route. Germanic migratory tribes would replace the ruling elite and their vernacular language and their body of laws. A prime example would be Gallia, becoming France due to the presence of the Franks or Spain, becoming first Visigothic and then Islamic after the conquest of the Umayyad caliphate.

I hope this answers your questions, even if not completely.

This is a question that comes up regularly here on AskHistorians.

/u/Daeres has previously answered Why do we call Italians Italians? Why aren't they called Romans [anymore]?

/u/bitparity has previously answered Did Italians in the Early Middle Ages still think of themselves as Roman?