I have a desktop and a laptop, so if I were to buy Skyrim, could I install it and play it on multiple computers without any trouble? Thanks in advance for any help.

If you buy it on Steam you can install it on as many computers as you like, as often as you like - but it does need to do a check when you start the game to make sure 'you' aren't playing it on two computers at once . You can play without an internet connection if you put Steam into offline mode, and for me (so far) if I start Steam with no internet connection it drops into offline mode automatically. I haven't heard that Skyrim has any DRM except for Steam.

Not sure about the boxed retail version, but I think it's the same.

It is, Steam is required for both.