Will I be able to transfer my gamertag to the new 360? I really don't want to get all my achievements again. I'm not a big fan of the 360, so I don't know if I can. If I can't, I probably won't get one. But I really want to play Gears 3 and Halo 4, and I miss having all the systems so I can get any game I want, regardless if it's a system exclusive, so I do want a new one. Also, if I can transfer my gamertag, will I be able to re-download all the stuff I bought, like DLC and games?
Yes, you can transfer your gamer tag and downloads. I recently upgraded from a launch console to a Gears of War 3 Sim model. Took about thirty minutes to transfer all of my save files, DLC, and such via a transfer cable.
Your Gamertag should still be no problem. I'm not completely sure about your DLC and such, but I think you'll recover your access to it without having to pay for it all over again once you recover your Gamertag. You'd just have to import your Gamertag to your new console in your case, which is fairly simple (you can find out how somewhere on Xbox.com). As long as you remember your log-in information for your Xbox Live account, you shouldn't have any problems.