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.
Quoted from http://support.microsoft.com/kb/906502
Important When you format a storage device, all of the information
saved on that device is erased and cannot be recovered. Make sure that
you either do not want the information stored on that device or have
backed up the information to another device before you continue.
To format a storage device
- Start your Xbox 360 console without a disc
in the tray.
- Go to the Settings hub, and choose System.
- Select Storage.
- Highlight the storage device that you want to format, and
then press the Y button on your controller.
In this case you want to select your Xbox Hard Drive
- Select Format. You receive the following message:
This will delete all content on this device. Do you want to continue?
- Choose Yes to format the storage device.
- If you are asked to enter your console serial number, enter your number, and
then select Done. This step helps guard against accidental formatting.
The console serial number is located on the back of the console and
behind the oval door on the front of your console.
What this won't delete
As mentioned above, this will clear the Hard Drive, but not your Xbox Live profile, which is stored on the Xbox Live servers. You will be able to use your profile on another Xbox if you sign in using your username & password.
What this will delete
All games, music, video, personalized settings and save games (unless they were previously transferred to another drive or the cloud.
Best Answer
Short answer: No.
When you purchase anything on the Xbox marketplace, you are purchasing it for your account. Even if you did not have the DLC on the hard drive, you may freely re-download them, providing your signed in to the same account.
That said, you state that you intend to set up a new profile. Your new profile would not have these DLCs "authenticated" to it, so they would not work.
Your best bet is to simply use the old Xbox profile.