TC, does Kotor 1 still not work? GOG labeled it as compatible with Windows 10 today, so you may have to re-download to see if it works. And what about KOTOR 2?
When I tried it today myself, it still wasn't working.
Compatibility modes on/off
Forcing one core on/off
Changing settings related to shades or FBO on/off
Opening videos there/removed
AMD dll fix installed/uninstalled
Widescreen mod used and unused
All of these were tried on a fresh win10 install of KotOR 1 which previously worked fine on win 7 and 8.1 on the same hardware. Each test was done by itself and then eventually combined with others to see if results improved.
Win 10 Pro x64, 4770k, HD7950 with latest drivers.
EDIT: Also figured I'd throw out there, but the start times for the game are massive, too. They take close to five minutes for anything of the game to actually show up on screen, even if it's just a black screen before the eventual crash message.
Pretty much everything Lurtz stated. The only thing I can think of is that GOG applied some kind of fix that screws with the game on Win10 in some way.