Yes, your girlfriend needs a visa or residence permit to stay longer than 3 months in Italy. The exact limit is in fact 90 days in any 180-day period, which also answers your other question: After an uninterupted 90-day stay, your girlfriend has to stay 90 days outside of the Schengen area (not just Italy) before visting again. Incidentally, it's 90 days and not three months, if she entered on July 1st, she has to leave by September 28th, not September 30th.
It is possible for the unmarried partner of an EU citizen to get a residence permit but you will have to prove you have a long-term relationship. I don't know the exact requirements in Italy but that could mean something like showing you have lived together and shared bills for a couple of years, not merely declaring that she is your girlfriend. Otherwise, she would have to qualify for a visa like any other third country (i.e. non-EU) national, whether for work, studies, or any other purpose provided by Italian law.