Probably the easiest way is to just buy a memory card and transfer things to it. The videos might be a bit difficult though, and you might just be SOL on that one. I'd check the Xbox website FAQ to see if there's any suggestions on how to do this. I'm guessing, knowing Microsoft, that the answer is going to be basically, "Too bad."
There are some third party attachments that will let your store data from your 360 onto your computer, but I can't say how they would affect the warranty on any new unit you may buy.
I do share in your pain. I'm preparing to ship my 360 back to Microsoft since the optical drive is on its last legs. It can take me over a half hour of opening and closing the tray before I can FINALLY get it to recognize there's a game disc in the drive. It's a real shame that out of the current gen of consoles, the 360 has the bulk of the games I want to play, because Microsoft has continually screwed up the hardware end of things with the 360. RRoD, loud optical drives, optical drives that scratch discs, overheating video chips, optical drives that give out in under a year, the E74 issue... You can do absolutely everything right, baby the system to no end, and it will still crap out on you.
I have a Halo 3 edition Xbox 360 that is acting up, I think I might get RRoD soon. I want to purchase a new one but I want to keep all my gamesaves and purchased videos and such and install into the new 360. Does anyone know what I need to do this? Please advise.