I wrote gog.com an email about this issue with win 10. They did respond positive. So they might actually fix it. Stay tuned!
Cool. Because I tried to contact the support and the story so far is - after going through the common stuff like re-installing the game, checking the admin rights, trying another PC, etc, they've just offered me a refund (I'm not going for it). Meanwhile, I keep trying things... Can't find the actual thread, but somebody here, on GOG forums, said that the original CD version of Dark Reign works fine on Win 10 and it's something related to GOG's "no CD" solution. After hours of looking, I found my good old CD (I have only the original game, no expansion). Surprisingly, installs fine, but asks for CD every time I'm trying to start the game (clearly, copy protection doesn't work on Win 10). I tried some different "no CD" .exes from the net and I tried the unofficial patches. 1.4 and 1.7. I also tried your fixed .exes from the other thread here. All of those help with starting the campaign. But neither of them help with the crashing problem. In other words, my old CD version acts exactly the same with the one from GOG. Maybe it's because GOG version left some traces in my system? I doubt it, though. Since usually when you select "uninstall all componentes", GOG uninstaller does delete everything. So... yeah. The problem stays. And the worst thing is... I can't even guess what's the source of it. Tactics.log says "ERROR Save 1058". But WHAT IS that "ERROR Save 1058"?! T_T