HoMM 3 Complete will not install on my Windows 7 64-Bit system. What the hell?

What error messages, if any, are you getting? Did you run the installer as an administrator? Where did you install the game? Did you contact Support?

I've installed Homm 3 Complete from GOG on Windows 64 so the problem is with your system.

I'm curious about it as well since I use a WIndows 7 64 bit system and considering buying the Heroes 3 gog version.

Heroes 3 works just fine on Win7 64, except for the (easily fixed) multi-core crash. You can either set the core affinity manually each time you run the program via Task Manager... or you can permanently fix the problem by using imagecfg to set the .exe to use only one core. Details on imagecfg are here. That bug affects all versions of Heroes 3, though, not just GOG's.