Old Hard Drive on new XBox
Solution 1:
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.
Solution 2:
You definitely want to move your profile over before removing the HDD from the old enclosure. This is most easily done with the XBox 360 Transfer Cable. Otherwise, old product keys (for DLC, XBLA purchases, etc) might not work properly with the new 360.
As for your second question, I would imagine you could put anything on the internal storage that you wanted, as long as it fits.