Italy is part of the Schengen Area. For short stays (not exceeding 90 days in total during any 180-day period), the whole 26-state area is treated as a single country as regards passport validity, maximum visa-free stay and passport stamps.
Your passport must remain valid for at least three months beyond the end of your proposed stay. If only planning a very short visit, a little more than three months left would be fine; for the maximal 90 days of visa free stay, you would need about six months. You have more than that left on your passport.
If all relevant documents match, most importantly booking/tickets and passport, it should not matter whether you use your present or your former name. Otherwise you should carry some sort of proof of your name change.