AZ lounges are pretty bad - skip them. Not sure what else is at JFK T1, but Priority Pass lists KAL Business Lounge. I've never been there, so I can't weigh in.

Carry on if you can pack that way. It's much easier and since you're in J, there should be plenty of bin space.

I think your 25-35 min connection to the train is a little tight. Too much can go wrong - the Leonardo can be late, immigration can take a while, whatever. I'd book an hour minimum. Service to Florence is frequent.

FWIW, on the Frecciarossa, we've always booked 1st class, but 2nd class is actually not much different and it's not a very long ride - 2 hrs or so.