FIOS has a considerably faster upload speed, and when it comes to picture quality on the SlingBox, that is pretty much all that counts. So, yes, FIOS would give you a better picture most likely.

The best way to set up your SlingBox would be to connect it to your home DVR (Cable company's, a Tivo, etc). This way all your shows are in one place and you don't have to always have your computer on to record. You just simply log into your SlingBox from the road and use the virtual remote to control the DVR next to your TV.