I have tried so many ways to get Stronghold to run, but my best guess is that since I upgraded from Windows 7 to Windows 10, something happened, that the team at GOG has been yet unable to help me figure out. I have tried running the game with all other applications turned off, and in safe mode with the same error message. I have completely removed and reinstalled several times. I have tried all previous versions of Stronghold --- same DLL error.

Faulting application name: Stronghold.EXE, version: 0.0.0.0, time stamp: 0x5aac04ca
Faulting module name: igdumdim32.dll

Stronghold is searching for a file that isn't in the directory. The igdumdim32.dll seems to be an Intel Graphics file. The 32 file isn't there, only the igdumdim64 one is.

The only thing that has worked is when I roll back my Display driver to a previous version. Everything is skewed, bright, and ugly, but Stronghold HD and Crusader work. But when I update the Display driver, and try the game I get the same DLL error message again. My best guess is that my OS previously was 32-bit and now it is 64-bit. But why would a fresh install still not work? Does anyone have a similar error or some insight?

Have you tried setting the game to run in compatibility mode for Windows XP? Read here.

Yes, many times, with fresh installs of GOG galaxy and the game itself; still received the error message above.

The link I posted also suggest to set the game to run as administrator and enable DirectPlay.

I have tried both multiple times and again after fresh installations, and it does not fix the DLL error message I am receiving.

I'm afraid I can't help you any further, sorry.
If you don't get any help here, I suggest you to contact GOG support or the developers.