A normally formatted memory stick (not one for save files) can be used to play music/videos etc. stored on it. You can't copy files to the Xbox, but you can use them from it.
You can't add to the Xbox formatted filesystem from a computer.
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.
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As far as downloaded digital games are concerned, one just needs to plug in a USB drive and copy/move the file from the internal HDD of the Xbox 360. Then you one just needs plug in that same USB drive to a Windows PC (tested on Win. 7) and just copy and paste the files to the desired backup location.
The formatting was found to be,
where the Title ID correspond to either the main games or XBLA games
As far as games ripped from the disks are concerned, it was not possible to transfer from the 360's HDD to the USB drive. Its likely that the disk has to rip directly to the USB drive in the first place for this kind of backup to work.