hostgator only copy your forum over to their servers, you have to point your domain name over to your new server. i had a few problems with my board as i ran php5 on my old server and they put me on a php4 server, i got switched over with no problem, even got an appology for their mistake, cant fault their customer service, even during the night to them as i'm 7 hours ahead lol

Yes if both servers are running some form of a Control Panel (cPanel, Plesk, etc) You are usually able to just transfer using the control panel software the entire account.

if you can get shell access on your old shared server, that makes it a LOT easier, since you can zip up your whole webroot and only have to download one file.

^which is rare.. not alot of Shared hosts give out SSH access but still, he's right

Yes. I agree. Find out if you can get your site hosted on a server with the same type of control panel and they will import it very easily.