After hitting "Play" on the initial options window, the screen goes black, a window pops up in low resolution and then I am returned to the Galaxy screen. I get no message and it would appear as if the game ended normally. I am running Galaxy on Windows 10.

Try launching the game outside Galaxy, using its own executable.
Setting it to run in compatibility mode for Windows XP might help.

Tried that. Now it crashes at random times before the game begins.

Does not matter what I do the game just crashes on start up. Load, choose resolution, black screen, crash.

:(

Figured it out. I was setting the launcher and not the game exe.

Locate the "Demonstone.exe" file, not the launcher that the game uses by default.

Mine was in "E:\GOG Galaxy\Games\Demon Stone\Demonstone.exe"

Right click on it and select Properties. Then select the Compatibility tab, click Change Settings for all users and change the compatibility mode from Windows 98/ME to Windows XP (Service Pack 2). OK back out of the dialogues and then launch the game via the Demonstone.exe and NOT the launcher.