Description of problem:
During gameplay, things appear fine, but in menus and cut scenes, the content on screen appears to be shifted upwards. The bottom half of the screen appears black, and the upper half of the screen shows what should be on the bottom half. Menus are still somewhat navigable by guessing what the upper half is supposed to look like, but it really ruins the fun in cut scenes.

That's a new one, not heard of anyone else having that kind of issue. I'd say it is best to raise this to GOG support.

What resolution screen do you have and what resolution is your nGlide set up to use? Could it be something there causing this?

My monitors default resolution is 2560x1440.
I tested the game with various resolutions in both Glide and D3D mode, the problem is still the same.

Okay, well that's a standard 16:9 so that's okay, but it is a very high resolution. I'd set the game up to use 3DFX as it is from the default GOG installation. Then open up the nGlide configurator and set the resolution to your desktop. Then try playing. If it crashes or doesn't look right, change the resolution in nGlide configurator to a smaller 16:9 resolution.

Note that there is a ratio option in the nGlide configurator. This just affects the area of the screen on which the display is output, and not your screen or resolution aspect ratio. I keep it at 4:3 so that the game is pillarboxed to appear how it was designed to and isn't stretched, but you can use any option, it is personal preference.

If this still doesn't work, I'd definitely contact GOG support (my experience of the D3D and software modes is that both are very buggy and barely work). You could always try replacing nGlide with dgVoodoo if you have a DX11 GPU, but I don't know how that works as my GPU is only DX10.

I tried adjusting this as you said, but it appears to be only affecting resolution during gameplay. Menus and Cutscenes appear to override this setting, and it is there that half of the screen goes missing.

I have now also forwarded the issue to GOG Support as you suggested.

Hmm, interesting. Hopefully GOG will be able to help. Something else you can try, though, that I only just found out about, is a specific DDraw wrapper for Outlaws. See this post made not long ago:

http://www.gog.com/forum/outlaws_a_handful_of_missions/directdraw_wrapper_more_pixels_and_full_hud/

That might allow you to play in D3D mode. I'm going to give it a try myself a little later.

I tried the Direct Draw wrapper, posted by n0th.

If you can believe it, this did actually solve the "half screen" issue. More precisely it appears to be linked with the "NoVideoMemory" option in the aqrit.cfg that comes with it. When I enable that, the menus and cut scenes are whole again.
...However...
New problems arise with this, such as game crashing when re-entering the menu after being in game. This basically makes the game uncompletable, because I cannot save the game. Also, I am not a big fan on the low resolution (in case my monitor was no indication) that his solution appears (intentionally) limited to. :(

Yeah, I just tried that DDraw wrapper myself and had the exact same issue with the menu not re-opening after being in game. I also had a very long pause between the opening cinematic and the main menu. I'll be sticking with nGlide.

I wonder if any of the other nGlide settings or the advanced settings in the Outlaws config utility would solve it for you? Have you tried using the resolution by app option in nGlide? Though this will then run it at a lower resolution like 800x600 or 640x480, which isn't really what you want.