[circonian]

I do not use a multiple monitor set up so I can not test this myself.

But did you realize if you, as an example, open your inventory to get in UI mode move your mouse (anywhere you want) then exit UI mode (hit esc, move, whatever). Your mouse cursor will stay in its last position. Next time you enter UI mode the mouse will be where you left it.

My point is, what would happen if you left it off of the screen on another monitor that is not running the game? Would this cause a problem?
EDIT: Oh, I didn't read everything...apparently it sounds like the answer is yes from what sirinsidiator said.