would it be legal if I were to name my band "Vault 101?"

Yes, since the game itself is not trademarked under the name Vault 101, and the fact that band names can be named the same as any other act/product, besides other musical acts. For example, if you named your band Forrest Gump, the movie studio couldn't sue you, but if there was already a band out there named Forrest Gump, they could. There is a band from my hometown called Stretch Armstrong, they informed me about all this.

Well, say that the trademarks were outdated, then yes you could. But now i do not think you could as you could get sued. Im 98 percent sure that if your band is ONLY for home use that you will get in NO Trouble at all. ONLY IF IT IS FOR YOUR HOME VIEWING! Otherwise no. You could now.

Ironically, you could name it "Fallout."

I've heard of a band (vaguely) called Sauron and one called Minas Morgul.

In band Vault 101, does anyone enter and never leave?

Vault 101 is not a registered trademark, it's just a text description followed by a number. So no one could stop you from using it in some other context outside the game.

I'm a musician as well, my advice, just do it. You should'nt have any problems, none you would have to worry about until you get signed anyway. My stepson is calling his band gorebag.