I get the message "This application failed to initialize properly (0xc0150002)" immediately upon running the .exe. I can open to editor just fine and the installer checked the file integrity check. Trying to run it in Compatibility mode for windows 95 and 98 both failed. I'm running it in WinXP 32bit.

This is something I found relating to your error code, though the resolution is for another game.
http://www.crimecraft.com/forums/Hardware-and-Software-Issue/topics/application-failed-to-initialize-properly-0xc0150002?page=1#post-99791
Essentially, make sure you have installed Microsoft Visual C++ 2005 SP1 Redistributable Package and have your computer up to date.

I have the same error. Make sure you submit it to GOG's support, so that they can look for similarities in dxdiag output, etc.

Either of the following might work:
Install Visual C++ 2008 Redistributable
Install .NET Framework 3.5 Redistributable
Clean your Registry

Installing .NET 3.5 did the trick. The game works fine now. Thanks!

What the...? Is this Stardock's fault, or GOG's? Because Total Annihilation sure as hell didn't make use of .NET anything.
Does anyone know whether it actually requires .NET, or might it be a dependency on some other library that happens to ship with .NET?

After looking at all of the DLLs loaded for TotalA.exe on my Win 7 machine, the dependency is probably tangential. The graphics and sound libraries, for instance, will still depend on the drivers you have for your current hardware, which may in turn make use of more recent libraries.

Thank you, folks!
The suggestions had seemed a little too vague/optimistic for my liking, but I got tired of waiting for any kind of acknowledgement/response from GOG's support facility (especially after skimming through this thread), and looked up the Visual C++ 2008 Redistributable, which sounded much more palatable than the comparatively monstrous .NET package.
It's a 1.7MB download and I'm happy to report that it did the trick for me. The game now runs fine!
Much appreciated.

I have the exact same problem, thank you for the suggestions, I will try them.

EDIT: the visual c++ redistributable worked for me.

Many thanks for the advice. I had no idea why it was not running while my cd install was running fine. The visual C++ redistibutable solved this problem. Much appreciated.

Thanks for the solution!
Installing Visual C++ 2008 Redistributable did the trick!
I have TA running on WinXP with Virtualbox.