if you haven't booked the flights yet, just book with an airline that will check you and the baggage through, regardless of a bit higher price, it is worth as it means less hassles
One solution is: fly without checked in baggage. As you can easily - showing your ticket to Italy - proceed in BKK straight to the Transit zone, even with two separate tickets
the visa exempt is called a "visa-less" entry. Italian passport holders will receive a 30-days "admitted stay" stamp in their passport at the Immigration point. You COULD be asked to show the "three proofs" but it is highly unlikely, and in case IF you should be asked, you can show them the ticket out to Italy on the same day.
you do not need a visa for Thailand, for a transit through Suvarnabhumi Airport to Italy. You but would need one, if you arrive at Don Muang and needed a transfer to the Suvarnabhumi airport. Just make sure you arrive and fly out of the same airport