Yes, they are both correct. In the first example, "used to" suggests that you were engaged in a process of learning during your time in Rome: e.g. by going to classes on a regular basis. The second example suggests that the process was probably 'completed' at the time you were in Rome, and that you have now mastered Italian.

Both sentences are correct and their meaning is clear, but only #2 feels natural. #1) The combination of "I used to" with "while I was" feels awkward because these two phrases compete with each other to create the extended time frame. Use one or the other: "I used to study Italian WHEN I was working in Rome." "I STUDIED Italian while I was working in Rome"

Yes, both sentences are grammatically correct. They convey the same meaning, but they use different verb tenses. The first sentence, "I used to learn Italian while I was working in Rome," uses the past progressive tense ("was working") and the past simple tense ("used to learn") to indicate that learning Italian was a regular or habitual action in the past. The second sentence, "I learnt Italian while I was working in Rome," uses the past simple tense ("learnt") to indicate a completed action in the past. It suggests that you finished learning Italian during the time you were working in Rome. Both forms are commonly used, and the choice between them depends on the specific context and emphasis you want to convey.