My area of expertise here is after the fall of the Western Empire so I'll answer your second question. Keep in mind that this topic is controversial in Italy today and many historians have a bias.

To understand how Roman identity in Italy changed after 476 AD, you need to understand the history behind it. When Rome "fell" what really happened was that a Germanic warlord named Odoacer captured the boy emperor Romulus Augustulus in Ravenna. Odoacer was not an invader however, in fact he was an officer in the Roman military. His reason for attacking Ravenna was not to destroy the Roman Empire, but rather to gain the lands promised to him and his foederati (non-Roman soldiers) by Romulus' father. After exiling Romulus, who notably was not recognized by the Eastern Emperor, Odoacer proclaimed himself King of Italy. His kingdom didn't survive his death however, and the Eastern Emperor Zeno offered a deal to Theoderic, leader of the Ostrogoths. It stipulated that if Theoderic conquered Italy, Zeno would allow him to rule it as the representative of the Roman Emperor in the east. Theoderic was successful but in reality, the Eastern Emperors had no control over Theoderic or his successors.

By this point, Italy has seen many incursions by non-Latin peoples, even if many of them were technically Roman troops. However, Italy remained solidly Roman throughout the Gothic rule. The Ostrogoths allow the natives to continue on as though nothing had happened, maintaining their law and identity while some prominent Roman writers like Boethius even flourish during this time.

The ambitious emperor Justinian comes to the throne in Constantinople and uses the execution and deposition of an Ostrogothic queen as pretext to invade Italy. The subsequent Gothic War is devastating to Italy, as the Eastern Romans and Goths seesaw in control of the peninsula. By the end of the 19 year war in 554, Italy is in ruins. Most cities have been depopulated, the countryside destroyed, and Rome itself lacks the aqueducts that allowed it to support a population of up to a million people under Augustus. The Eastern Romans or Byzantines are in control of Italy, but that Italy is a shadow of its former self, and a Persian threat from the East causes most troops to be withdrawn from the peninsula.

This underpopulated, barely defended Italy is invaded by the Germanic Lombards just 14 years later, who manage to conquer the majority of the peninsula quickly, only lacking the Greek extreme south, Romagna, and Latium. Again, a Germanic people floods into Italian and like the previous times is not numerous enough to completely erase the native Romans. But this time, the native Romans are in a state of massive disrepair and their numbers are vastly lower than before the Gothic War. There is little resistance they or the Byzantine troops can offer against the Lombards.

Of course, Latin continued to be spoken and even the Lombards adopted it extremely quickly. The Lombards also adopted Roman dress and occasionally Roman names. Although they followed a Germanic Lombard law they allowed the Romans to follow their own. But in the lack of much culture after the devastation of the Gothic War, few institutions remained to support Roman culture and identity. Gradually, the two groups mixed and a hybrid culture emerged. Genetically, the Lombards did not much affect the Italian population, but the Italians eventually adopted Lombard laws, albeit much influenced by Roman ones. Lombard names became popular, and gradually, many people began to see themselves as citizens of a Lombard Kingdom, subjects of the Lombard King. We can't really say how most common people identified or what term they called themselves, Lombard or Roman, but it seems that whatever the case, the fusion of cultures was one of the primary steps in forming an Italian people.