If I try to run Heroes II, the WIndows version available under extras, in full screen mode, it crashes the game in such a way that I have to uninstall it in order to play it again. This, obviously, sucks, and I'm wondering if anyone else has had this problem here on GOG and knows a way around it. Likewise, it won't let me change sound options off MIDI, saying my CD drive is unavailable or incompatible.

Why not use the Gold version, then?
It has the added benefit of looking better (image not stretched) on a modern monitor if you use the Keep Correct Aspect Ratio function of DOSBox.

If you want the honest answer, because I want to cheat. Specifically, I want to be able to use the debug menu, for options like unlimited movement, or have ghosts in your army, that sort of thing.

Have you tried the "obvious" solution? Different compatibility modes, Run as Admin?

Can u send me a link how to use the debug menu? :)

Right click on the desktop shortcut icon of the game

Go to Properties

Find the Line that says Target. it's the file path to the game program

Go to the line, and then to the end of it, past the quotation marks, and put in a space and then /nwc after it.

Use that shortcut to run the game. In windowed mode, you'll see the typical tool bar at the top, one you're in a map, the tool bar will have the debug option on it. This includes giving you 999 mana and all spells. Unlimited movement, and adding creatures to your army, as well as secondary skills and resources. Note, to use most of these, you need to select the hero you want to use them on.