Well, this question honestly begs a very complex answer.

Of course, the legacy of Rome was probably stronger in Italy moreso than anywhere else. Each city saw itself differently in relation to Rome and this view changed over time. Florence, for example, saw itself as a daughter of Rome. The original settlement is very likely of Roman origin, but various chronicles played this history up and expanded on it. Genoa, likewise, developed a complex origin story based on Roman foundations. Jacopo da Varagine's history of the city begins with the city being founded by Janus, the first king of Italy (mythical, I'm not sure of the source of this myth) and its enlargement by a second Janus from Troy, placing Genoa not only within a Roman narrative, but almost contemporary with the great city. Venice had a different conception of its "Romanness." For hundreds of years, even into the end of the 12th century (I believe until they sacked the place) they asked the Emperor of Rome in Constantinople for the right to rule. This is a completely different conceptualization of the Roman tradition.

This connection to Rome was embellished during the Renaissance when Roman culture and traditions were very much in vogue (referring to the Pope as the Pontifex Maximus is actually a Renaissance conceit, for example).

All this begs the question, how much did your everyday medieval Italian care about this connection? My answer is probably not much. Medieval Italians developed intense, local identities that intertwined not only the Roman past, but especially ecclesiastical history. The best book on how medieval Italian cities constructed identities, for my money, is Augustine Thompson's Cities of God and there's not a lot of Rome (outside of a Christian sense) in that book.

That's a difficult question, not in the least because Italian urban identities and post-Roman ideas of identity are separated by a few hundred years.

Prior to the Italian Wars of the sixteenth century, we don't get many broad exonyms and endonyms in Italy. In the enormous window of time between the seventh and fourteenth century, Northern Italians are sometimes lumped together as "Lombards" by foreigners. But you also get a fair number of gray zones where the term stops getting applied as early as the eleventh century, especially in large border regions like Piedmont and the Friuli. Likewise, areas with strong local identities, like Rome and Venice, seldom receive the "Lombard" exonym (in the case of Rome, they were never lumped with the Lombards). The South is another world altogether: documents, notably Venetian laws regulating trade, by and large grant exonyms only to specific ports. When they do need to generalize, the Venetians normally use the demonym of the southern monarchy's court: this leads them to calls southerners "Beneventini" until the eleventh century, subsequently switching to "Siciliani." However, these terms are very rare, almost ad-hoc, and the people living in the Southern Italian mainland would certainly not take kindly to being lumped with Sicilians. In fact, starting from the ninth and tenth centuries, by and large local identities were the strongest kinds of identity all over Italy. The ties between these local identities and the Roman heritage are complex.

First, consider the most recent meaning of "Roman" for Medieval Italians: Between the end of Justinian's reconquest in the sixth century and the Carolingian conquest in the ninth century, Italians tended to identify themselves as either "Lombard" or "Roman;" but the word "Roman" oscillated between very limited meanings, denoting either the shrinking military outposts of the Byzantine Empire and actual residents of the city of Rome. As early as the seventh century, the both the Lombard Kingdom and the Byzantine Empire begin using the words "Lombard" and "Italian" interchangeably, while the "Romans" are the people on the Byzantine outposts in Rome and Ravenna with whom the Italo-Lombards are locked in systemic conflict. In short, according to seventh and eight century Italians, Romans were "the other." If in the late sixth century King Agilulf and queen Teodolinda had advisors named Paolo, Pietro, Stabliciano, Secondo, Aureo, Pompeo, by the next generation King Arioald's advisors were named Adruvald, Rodoald, and Ilbichis; all native speakers of the Latin Vulgate, no less. Clearly, the Lombard and Roman segments of society had fused. The product of this fusion? Italy.

Given all that, it's understandable how following Charlemagne's conquest he crowns himself "King of Italy;" and the title "Holy Roman Empire" would only be contrived two centuries later. Although it's certainly fair to say that in the following centuries, Medieval Italians certainly were aware of Roman history and must have felt some of it as their own: Dante fills his Divine Comedy with heroes of antiquity, after all. But it is extremely telling that when his protagonist meets the Roman poet Virgil, his guide in the Inferno, his introduction reads "My parents were Lombards, Mantuans by homeland both."

Dante has Virgil, possibly the "most Roman" writer ever, introduce himself as a Lombard.