When the window is in the lower or right part of the screen and the resolution is reduced so much the window disappears and the only way to see it is by maximizing. Is it possible to keep at least part of the window on the screen in this case?

I don't see a way how to do that. It looks like a problem of a window system you use (OS bug).

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I'm using windows 7 and I've never seen another program hide it's window in this situation. If it helps, here's some java programs that keep the window when reducing resolution:

Both programs at Twan's Tools
BDSup2Sub with source code