The Arks were created as a competitor to Vault-Tec's Vaults, which means that Ark people and Vault dwellers are both a sort of stranger to the post-apocalyptic world. As for the difference in apocalypses, it's a result of unreliable stories. Some thought it was a meteor while others thought it was a nuclear war. Feltrite could supposedly be a form of irradiated crystals.
The Authority's flag is so obviously a Nazi war flag that it has to be deliberate. Now that wouldn't make sense to use it as a symbol in our universe, where Nazism was the ideology behind Germany's twelve year tantrum across Europe which was defeated and destroyed. In Wolfenstein, where Germany conquers the world, Nazism is the ideology of a global superpower that did what no other country had down in terms of (albeit violently) unifying the human race, something General Cross may be trying to convey by using a modified version of their symbol. Furthermore, the advanced technology possessed by the pre-impact humanity seems advanced for the 2030s, so the technological advancement caused by the unearthing of Da'at Yichud vaults could explain that.
As for Doom, this is less serious, but we keep seeing the Doom 3 symbol painted alongside areas the Mutants infest. Is ID just reusing perfectly good artwork, or have the mutants become aware of hell and begun depicting the demons in their artwork?