Was the Fallout game series inspired by this?
Possibly.
The war in Fallout was because of dwindling resources and not so much absurd stupidity.
I think he's referring to the "live in underground shelters/society in vaults" idea, proposed by Dr.Strangelove.
I think this movie is much more like Metal Gear Solid (especially Peace Walker) than Fallout.
No, fallout was inspired from the Mad Max films I believe. Fallout 3 even had the same shot where Mel and the dog are walking down the road with their backs toward the camera.