If your arriving flight is on time and if you do not have a long line at immigration and if your checked bags are not held up, you will have lots of extra time to kill. (That's what happened to me when I had three hours to make a very similar change on separate tickets). Of course, if there is any delay in one or more of these steps, you could be rushing, or could miss the next flight.
Yes, you will arrive in T3, go through immigration, pick up your checked luggage, go to T1, and check in for your Alitalia flight. Since you can do online check in and print a boarding pass 24 hours before departure, you should be able to print your boarding pass for your Alitalia flight while still in the US.
Yes, it's only about a 10 minute walk between T3 and T1; we did it outside because the weather was nice, but I bet there's a way to do it indoors.
When I took them, Alitalia domestic flights included one checked bag and allowed the same sized carry on as my flight from the US to Rome. Of course, this was April 2014, so it could have changed.
If you haven't bought your Alitalia ticket yet, be sure to use their International website: https://www.alitalia.com/en_en/. This has the same prices as their Italian website but has an English option (the Italian one has Italian only). The US website has much higher prices for domestic flights in Italy - for the exact same flights!