If I'm having trouble with space on my computer, can I copy a movie to my iPhone/iPad/iPod Touch, delete it from my computer, and still keep the movie on my iDevice (even after another sync)? Or is it like apps, where I need the app on my computer if I want to keep it on my iDevice?
Yes, just select "manually manage music and videos" on the summary page in iTunes and the items on your device will no longer be tied to the iTunes library.