Just giving an update
I just got off the phone with both United and Lufthansa. They have room in-cabin on all flights so I made reservations for my doggy already and he will be with me in the plane the entire time. The price will be $105 each way (from U.S. to Italy and then Italy back to U.S.), and weight just has to be under 18 lbs (carrier included) and the dog must be in a carrier that will not leak (22L x 14W x 9H). Hope this is useful information to someone else as well