I've had my children in Italian schools (albeit private ones) since we arrived in Italy (this is our 4th school year). The best option is to do what you are doing and 'visit' schools that you may be interested in, check out the classrooms, etc.
I strongly recommend visiting the classroom your child would be assigned to see how the teacher behaves and how the students behave. Italian students, in our experience, tend to be a bit more aggressive and disruptive than Americans, but a good teacher will keep this in check. If not, you don't want your child attending.
There is no smoking allowed on school premises, but that doesn't mean that parents adhere to it. In fact, our current school just had to send out a notice to Italian parents reminding them that it is absolutely forbidden to smoke on school 'property' not just inside the school.
There are good Italian schools, you may just have to venture beyond Gricignano to find them. Don't give up!!!