>Is there a way to have Word to remember the last window position of the last document I use? I always find myself adjusting the window position as every time, most of the times I should say, Word is showing me the document in a position that is different than the last time I was inside Word.

It doesn't seem to do that to me unless the last time I shut it down was on my second monitor and the next startup (of Word) the second monitor is not enabled.
Is it possible that you have one instance running already (minimized)? In that case it will open the next offset from the other.