Has the GoG version of Shadow Magic been updated or otherwise fixed in any manner, so it'd work on Windows 7 (mine's 7) and possibly later operating systems?

I actually own the physical copy of SM myself, it installs too (usually, after some trouble), but no matter what I do I can't get it to run. Administrator rights, various compatibility modes, both official and fan-made patches; none if works. The game either flashes the logo and shut itself down right after, or just shuts down right away.

All gog games work on Windows 7. This is why we buy our good ol' games here. :-)

Shadow Magic definitely installs, though you may need to mess with the selections with the config tool. User patch and fanware all install without a hitch. This includes the email wrapper if you want to multiplayer (PBEM).

Very nice if so, just might shell out the cash for it.

Windows 7 is missing some older DirectX 9 components. If you've never installed them some older games don't start.
Use Microsoft's DirectX end-user redistributable to complete your system. Extract the files into a separate folder and run dxsetup.exe for installation.
http://www.microsoft.com/en-us/download/details.aspx?id=8109