Having just recently gone through something similar... (My 360 died and I had to get it fixed, they sent back a different 360)
1) Yes - you can recover your gamertag from XBL
2) I'm assuming you mean FFXI? When Microsoft released the bigger hard drive there was a hard drive transfer kit that they offered for free. I'm not sure you can do that with FFXI though. When I got my new unit and put my old drive on it, Playonline wouldn't start. So I had to reinstall the entire game. Guess the FFXI installation is tied to the Xbox hardware somehow.
3) Yes - deleting the FFXI from the hard drive wont have an effect on your account