Pressing Alt+R while in-game ups the resolution to double its default value. I don't think the d3d patch works with the GOG version, because it's the DOS one, not the Windows version.

If the patch includes the executable, it should, as long as you find a way to bypass copy protection.

The D3D version looks worse and removes some visual effects from the game. Honestly, don't bother with it.
If you're desperate though, the original executables (deepd3d.exe, keepd3d.exe) are available in this self-extracting archive.

The GOG version can swap from 320 x 200 to 640 x 400 by pressing ALT+R and it looks decent enough. If the game runs too slowly you will need to edit the dosboxDK.conf file, scroll down and change the line cycles=80000 to cycles=max

If you want high resolution and widescreen support you should use KeeperFX instead.