A Likely Solution for People Who Can't Get This Game to Run
Go into your Doom 3 BFG Edition folder that's in your steamapps location, find the folder msvc_2010, open it up, and run the vcredist_x86 manually. Usually the vcredist_x86 is automatically run whenever a new game is launched for the first time through Steam, but for some reason, Doom 3 BFG Edition doesn't do this.
Customer support is completely incompetent, telling people their usual default run of "delete the clientregistry.blob and check your router" which just doesn't work. I even told them EXPLICITLY "I don't even get the standard first-time-run C++ Redistributable window." There's something about Doom 3 BFG Edition's programming that doesn't run the vcredist_x86 program on its first run, and so that might make it unplayable. (For some reason, I could run Doom 3 BFG Edition on one computer despite this problem, but that computer also had normal Doom 3 installed once upon a time.)
Maybe I should apply to work for Valve as customer support.