In the middle ages, most historians agree that the majority of 'Italians' at that time didn't consider themselves to be Roman. That ended around 700-800 A.D. when the Lombards (a tribe from the north, above the Italian peninsula) invaded and captured about 2/3 of Italy. The Germans to the north claimed to be Romans through the Holy Roman Empire and Charlemagne, and the Greeks to the east claimed to be Romans through the Byzantine Empire, an old relic of the Roman empire that had grown quite far from what it once was. These two big groups of people claiming to be Romans created quite an identity crisis for the Italians who, eventually settled on a regional Identity partially forged from the germanic tribes that invaded and the culture of the Roman populace before.

The craze in Europe at that time, at least among the nobility of Europe, was to be seen as Roman, not unlike how people in Rome and Carthage in ancient times were all trying to act like Greeks. Roman architecture, food and drink, Latin, etc... all were a big impact on society. Even then, at the time of the Renaissance and Middle Ages, people didn't hesitate to reuse, improve, or remodel older buildings.

*'A northern visitor to Rome about the beginning of the 13th century has recorded the vivid picture left in his memory by the first view of Rome as he caine by the road from the North over Monte Mario. And the two things that struck his eye were the great masses of ancient buildin’-s (nalatia, the Coliseum etc.), standing up smenr the houses or dominating the empty spaces within the walls and. the 9seges turriinn’, the forest of church end baronial towers. A number of the tall medieval camnaniles or bel1-towers still remain beside their churches, but they are exceutions, and far more have vanished in the reconstructions and alterationssince the Renaissance. Still rarer are the towers elonging to the fortified houses or castles of the great medieval fauilies, for the Renaissance palace has replacel the baronial castle. In the case of one of the largest roman ralaces, still belonging to one of the oldest Roman families, the Colonna, a medieval tower is even now attached to the Renaissance building.'

• G.MC.N.Rushforth, Pegasus Magazine 1985, University of Exeter*

In the Dark ages, Rome underwent harsh times and throughout, a lot of the old remaining culture and traditions carried out by the populace were stamped out by famine and conflict.

*'The relative prosperity of the days of Hadrian and Paschalis died away, and the 10th end 11th centuries are the darkest and most degraded period of Rome end the Papacy. Hitherto what culture and art existed was the last faint tradition of the art and culture of the ancient world. But no new life was put into it, and under the terrible conditions of the age the effort grew fainter, and the results were more and more degraded. In the some way the ancient city of Borne in its outward aspect was still there in the 9th century, with the old lines of paved streets, and the temples and baths and palaces, still containing some of their precious contents, but gradually fallinq into ruin; while all the church building and church restoration by the Pores was only carried on by using ur old materials and adantin’ old buildings. There was no fresh architectural or artistic movement.'

• G.MC.N.Rushforth, Pegasus Magazine 1985, University of Exeter*

The most information most peasants and commonfolk would have about the Roman empire was from the Bible with different parables and stories taking place in the old empire. On the other hand, for the nobility, they would have more information on history, including family trees going far back, sometimes (more often than not, dubiously) tracing back to Roman times.

The black death ravaged the peninsula, along with the rest of Europe, also leaving quite the brain drain, not excluding knowledge and passed down stories on the old empire. In addition to this, the decimation of small towns and communities in the countryside left a lot of young people to move to the cities to escape the depopulating countryside, causing a stronger need for renovation and removal of some of the older relics of the Roman Empire, furthering the removal of the history of the city from the minds of it's inhabitants.

A lot of factors concerning war, religion, illiteracy of the common folk, etc... had a huge impact on people knowing about Rome, not to mention caring about it. One could argue it wasn't until the period of Romanticism in the 1800's where people really started becoming interested in Rome, and looking back at it with rosy eyes. A smaller but not insignificant factor in the push for Italian unification.

This is a very difficult question to answer. The "medieval period" covers a huge stretch to time. I think at the very least it is nearly impossible to pin down exactly when Italians stopped thinking of themselves as Roman. Even after Romulus Augustulus abdicated the Imperial throne in 476, handing power in the peninsula over to Odoacer (Odovacer) as the new King of Italy, Rex Italiae, certain Roman structures remained in place, even as mere reflections. Boethius, for example, was made a senator under Theodoric. Peter Llewellyn produced a very readable history of Rome during the early Middle Ages called Rome in the Dark Ages. In it he describes how the "red socks" of the senatorial class were still worn, but they were worn by the clergy.

Of course, it is important to recognize that at some point, "old Rome" as I shall put it inelegantly, went away. But later rulers in the city of Rome attempted a renewal, a renovatio of the old order. We see non-noble inhabitants of Rome taking up the mantle of Republican Rome at various points. For example, in 1143, when the comune of Rome was established, the popolo established a "senate" to counteract the power of the Popes and the nobles of the city that was a self-conscious renovatio of the Ancient Roman senate. Quoting Chris Wickham's book on Medieval Rome:

Although Innocent [II] was absent from the city—from the Papareschi family of Trastevere—and a keen nepotist, tension mounted, particularly when the Romans won a difficult war with Tivoli in 1143 and Innocent made his own Peace. As Innocent was dying in the same year, the Roman populus created a Senate,based in the secuar political center of the Capitolio, and defended it against three successors until Eugenius III (1145–53) fully recognized it in 1149, inaugurating a decade of a balance of pwer between pope and Senate. In August of September 1144 the era of renovatio sacri senatus began, a uniquely precise and self-conscious date for a new commune, even if a year later than it might have been (a fact that has occaisioned much debate). The city's secular government henceforth focused on the Senate for over two hundred years: already by 1148–51 we have our first court cases heard by senators, and other senatorial institutions slowly came in across the rest of the century and after.

Of course, this new Senate was very different from its ancient counterpart, we should understand that these medieval Romans recognized and yearned for an imagined past where the populus controlled its political destiny and the Popes were not even a glimmer on the horizon.

Perhaps the most curious example of medieval Romans looking to renew antiquity is Cola di Rienzo. The son of two leather tanners, he refounded the Roman Republic in the fourteenth century, declaring himself a tribune of the plebs. Ronald Musto wrote an excellent book on the subject, and I would highly recommend it.

As far as Romans looking back and recognizing the previous achievements of antiquity, I cannot speak to that extensively. I can say that the population had contracted significantly, particularly in the era of the Gothic Wars and the Justinianic plague, and it never recovered. By the 12th century, when the anonymous Miribalia Urbis Romae was written, a popular guidebook for pilgrims, the forum had become a cow pasture. I would direct you to that book if you're curious how people understood the ruins of Rome. It's really not worthwhile for me to summarize, but suffice it to say, the author "invented" quite a bit about Roman history. Assuming you don't know Latin, there is an antiquated translation online: https://books.google.com/books?id=v86gAAAAMAAJ&pg=PP5&dq=Mirabilia+Urbis+Rom%C3%A6#v=onepage&q&f=false.

As for the rest of Italy, it's hard to make a generalized statement about their relationship to the past. One thing that should be noted, however, is that the structure of the Roman empire at its death was incredibly different from the Augustan era. For example, the role of bishops had become much more important, and the "Roman bishop" so to speak persisted into the medieval era in many ways. Local identity was much more closely tied to the local diocese and its continued evolution than anything else (see Thompson's Cities of God for more on this). Despite this, the high and late medieval Italian communes played around the Roman legacy in very interesting ways. Carrie Beneš wrote an excellent book called Urban Legends about how medieval Italian cities repurposed or adapted ancient Roman myths to tie to their identity. OFtentimes, cities tied themselves to ancient myths. Beneš describes how in thirteenth-century Siena, with it's Montelupo (Wolf Hill), built up an identity around its foundation by the twin sons of Remus who had been banished by the jealous Romulus. As another example, Genoa invented several legends about a mythical king Janus that stretched back to earlier times. Beneš doesn't talk about this extensively, but Florence, at various times, created an identity as a "daughter" of Rome, inventing histories of the city that stretched back to Republican times.

Even if much of these identities were created, it was a fertile seedbed for the interest in Romanness and Roman culture that accompanied the humanistic study of classical literature in the fourteenth and fifteenth centuries that we call the "Renaissance," but that's another kettle of fish.

We have to remember the “Middle Ages” cover a large period, roughly one thousand years. For my answer, I’ll mostly be addressing the late medieval period, i.e., the late thirteenth and early to mid-fourteenth century, and specifically through the lenses of Dante Alighieri and Francesco Petrarca, two of the so-called Tre Corone (Three Crowns) of medieval Italian literature.

Although it’s true the Byzantine and Holy Roman Empires may have been considered the spiritual successors to ancient Roman imperium, it’s also true many Italians —remember, the peninsula was not unified during the medieval period, and so there are differences between Florentines, Venetians, Sicilians (Normans), Genoese, etc. —recognized their ancestral connection to Rome.

In Dante’s Inferno, the first canticle of the Commedia, the poet describes a meeting between the pilgrim-poet and Brunetto Latini in Canto XV, in which the latter recalls the Roman founding and continued habitation of Fiorenza (Florence). Latini explicitly states how the “sacred seed” of the Romans lives on in contemporary Florentines (Inf. XV. 76).

Dante was also no stranger to Julius Caesar, whom he admired as perhaps the greatest secular ruler in history. These laudatory opinions of the poet are made further evident when he places both Brutus and Cassius in the lowest depths of Hell in Judecca, where the damned suffer for betraying their benefactors. Brutus and Cassius conspired against Caesar, and are found in two of the three mouths of Satan himself, joined by none other than Judas, who, of course, betrayed his benefactor, Jesus Christ.

Moving on to Petrarca, we know very well how the poet was enamored with the Latinate tradition, having written the majority of his works in Latin. This was, of course, common practice in the medieval period, but unlike Dante, who promoted the use of the volgare, or vernacular, Petrarca found greater solace in the language of Cicero and Quintilian. This being said, if we look at one of Petrarca’s vernacular works, Italia mia, the poet addresses his audience as “people of Latin blood,” right after he references the historical deeds of two well known ancient Romans, i.e., Marius and Caesar.

These are, of course, only two authors —albeit incredibly influential and well known in medieval and modern Italy —but the general idea stands: medieval Italians, even if only the educated elite, were well aware of their past and considered themselves to be the direct descendants of the ancient Romans and their sementa santa (Inf. XV, 76); an idea, which, unfortunately, also later influenced the twentieth century fascist ideologies of Benito Mussolini.