If I install Steam to an external drive, will it work when I plug it into other computers? Or does it depend on registry keys/some other computer-specific thing?
Steam will work on an external drive, and will run perfectly fine as long as you're not trying to simultaneously run it from the computer's internal drive.
Note: Even though installing on an external drive means the games / client themselves are on the external drive, you may wind up with your saves or DLC on the computer, rather than the external drive. (Exact effect depends on the game.)