I am not up on the most recent facts of working in Italy as a dancer but I know it was quite difficult back in the 1980s and I have not heard of it being on many lists of places for Americans to work in ballet. It had been easier as a teacher. Try reading the forum on Modern Ballet in Italy. There is a useful website as well as an audition listed in November.
Italy is a fabulous place to live and work. There are a zillion theatres most with Opera Companies and Orchestras with many foreigners working amongst the ranks however there are not that many ballet companies that work independently of the operas. Check Maggio Musicale Fiorentino, at Teatro Comunale in Florence. Also in Florence there is an independent but very good ballet company Balletto di Toscana but again for an American, I do not know. Also type in ballet in Italy and do a search. Perhaps you could choose a city and search that region, such as Tuscany, Reggio Emilia, Rome, Verona, Napoli, Milan etc. Buona fortuna, what a place to live and work!