Can I do a vanilla install of Morrowind on my desktop and another on my laptop, then mod up my desktop and copy the modded up game from my desktop to my laptop? If so can I then copy my save games back and forth so that I can play at home on my nice, comfy desktop and switch to the laptop while on the move?

Can see no reason why not, i do it without any problems

I haven't played on two different computers, but I did a similiar thing when I copied my mod list and the contents of my Data Files from one computer to a new one, so I didn't have to restart the save. It worked perfectly for me. I'd make sure I had Wrye Mash on one computer at least (both would be my suggestion), to be completely sure the load order is the same and to clean the save every now and again, but even without that, there shouldn't be any problems.