1) Fly to FUK, then. Much easier to get into town.
2) Yes
3) Send the extra luggage from airport to a Fukuoka hotel. Take the subway to Hakata and train(s) to Nagasaki. Stay in Fukuoka at the end for 2 or 1 nights. Buy a Hakata doll.
4) I wouldn't recommend any day trips if you stay in Nagasaki. You have multiple daytrip options from Fukuoka/Hakata.