If I plug the hard disk of one Xbox 360 into a different 360, will it pick up the user profile, all saved games etc. so I can pick up exactly where I left off?
You should not have problems. Since it is an HD that is "Microsoft Aproved", you are good to go.
There are some tutorials at the internet where some people are even using the old Xbox 360 HD inside Xbox 360 Slim version. Is just a matter of removing this drive from inside the "fast eject" case Microsoft created for its hard drives.
You can swap hard drivers on your Xbox devices, as long as they are using Microsoft certified hardware.
Yes, all save data, profiles, games, DLC, avatar items, game updates, profile pictures, and so on will carry-over. However, even though any physical data on the disk will transfer, some items will require a license transfer. Items that fall into this category are full-version games, DLC, purchased avatar items, purchased profile pictures, premium themes, or anything that was bought from the Xbox Live Marketplace. This is due to the license certificates being installed on the consoles built-in storage, and are not stored on external disks such as hard drives.
To do a license transfer, first ensure that anything you need to do this for is already on the hard drive. If you do a license transfer without the content being present, some if not all licenses for the content will not be installed. Now sign in to the profile that purchased the content and head to the settings tab on tbe Xbox 360 dashboard. Open the account tab and open the billing options page. All that's left is to select license transfer and you should be good to go.
To view and re-download purchased content from a profile, go to settings> account> download history.