So I tried every single little thing you suggested here, and not only does it still not work, but I had lost my saved games and will have to start over now because I didn't know that when originally deleting the unnamedsoldier.cfg would do that.
I did write down the error message it shows on the command line when I try to run the game with the 1600x1200 settings, and it says "GLW_StartOpenGL () - could not load OpenGL subsystem"
I tried the thing with the command line in game, and that doesn't work neither. I too don't get it because I play other games in that resolution and it's fine. Well, most other games, it's in 1600 x 900. I may be able to try it that way, but I highly doubt it'll work.
Sorry about that, your saved games aren't deleted they should be in the virtual store folder. (C:/Users/<your username>/AppData/Local/Virtual Store/Program Files/GoG.com/MOHAA)
This folder is also hidden so you need to show hidden files to see it. Copy and paste them back into your normal saved games folder. This a feature of UAC on Windows Vista/7/8. Windows doesn't let programs write to the program files folder anymore unless you run it as administrator. To get around this its best to install old programs into a folder outside of program files (eg C:/Games/).
With the OpenGL error its usually because it can't display that resolution. I'm not sure why your game can't display that resolution since other people seem to be running it in 1600 x 1200. You could try setting the resolution using the console in the game (press the ~ key ) using the same commands. Or your going to have to use a smaller resolution that does play.