In order to cancel an account, you have to call Xbox Support. I did that once after a trial period with Halo 2 on the original Xbox. They really tried to keep me on the service, but I suspect that won't be as annoying as you explain your situation to them.
I think I remember something about the gamertag you cancel being unavailable for a certain number of days or months in order to give you a chance to reactivate your account should you want to. That's probably automated too so I doubt they can make exceptions.
I know this is a late answer, so if you figured this out please tell us what you did as it's an interesting question!
First of all, keep in mind that on the 360, saved games and Xbox Live profiles are ultimately stored in different "places". While you can keep both on a hard drive (or on a memory unit or flash drive if the saved games are small enough), saved games are only kept locally, while your Xbox Live profile is tracked on Microsoft's servers.
If you're talking about an Xbox Live profile, then it will "move" the first time you recover it - once you recover it from Microsoft's servers, that becomes your up-to-date profile, so you won't need to copy it from your hard drive. Your saved games won't be accessible until you get that enclosure so you can connect your old hard drive.
Local profiles, ones not associated with an Xbox Live account, are only stored locally, so if that's what you have, then you'll need to wait for the enclosure to arrive.
Once the hard drive is connected to your new 360, you can move whatever saved files will fit to the internal flash drive. (If you go to My Xbox, System Settings on the far right, and choose Memory, it'll show up as Memory Unit (3.x GB free), and you can copy files from your hard drive to it and vice versa.) I don't personally believe there's a difference between storing your profile on the internal drive and storing it on the hard drive. However, if you find yourself playing away from home on occasion, you may be better off picking up a small USB flash drive (up to 16 GB), formatting it for the 360, and storing your profile on that. You can simply pull the USB drive and take it to your friend's house without needing to recover your profile at either place.
Best Answer
It appears that the gamertag is released back into the pool of available gamertags.
I changed mine a couple of months ago. Just now I logged into a different Xbox Live account and attempted to change that account's gamertag to my old gamertag, and I got the message that my old gamertag was available.
I also tried to send a friend request from this second account to my old gamertag, and I got the message "The gamertag you entered does not exist on Xbox LIVE" which further confirms this.
Microsoft is also releasing gamertags that were previously claimed and "dead" for one reason or another. They don't say why specifically, but I imagine gamertags associated with banned accounts are among those that may get auto-released.
As far as "aliasing" is concerned, when you change your gamertag, everyone who has you on their friends list will see your new gamertag in place of your old one, immediately. It's up to you to let them know that you have a new gamertag (or just watch them get confused when you play together, like I did... :P)