I like it! Did you come up with this yourself, or did you find it somewhere? Anyway, does extinction apply also to promoted pieces? Or only original ones? I mean, you promote a pawn and get a new queen, then lose your original queen. Do you lose the game (if you still retain your new queen)?
Extinction Chess
No, basically you only see the squares that your pieces can get to; everything else is hidden in some way, which makes a particular program for running it essential. It's not all that mindless, as you have to consider past moves and guess where your opponent might have moved, but for the most part it's sheer guesswork. Often leading to inadvertant sacrifices 
Yeah, I guess it has that similarity, except of course one can move pieces into the opposing half of the board as is always the case with chess. And, all you have to do is destroy the king, not the whole army.
I found this statement about the history of Extinction: "This game was invented by Games editor-in-chief R. Wayne Schmittberger in 1985 under the name Survival of the Species."
About the extra Queen scenario: you need to keep one alive on the board, and it doesn't matter which. You can have extra Kings as well. As I recall though, pawn promotion is rare -- if your position is dominant enough to get a pawn to the 8th, likely there were forced wins for you before that.
Is there any site that actually has an Extinction Chess server, rather than something that explains it?
There is a game site called "It's Your Turn" (http://www.itsyourturn.com/) that has Extinction and quite a few other chess variants. Most of chess.com's chess-friendly tools aren't available there, but basic mechanics were okay when I played there several years ago.
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I would love to see this variant on chess.com:
“Extinction” Chess Variant
1. The goal is to reach a position where your opponent is missing all pieces of one type: the King, or the Queen, or both Bishops, or both Knights, or both Rooks, or all 8 pawns.
2. All rules of orthodox chess are followed, with the following exceptions:
3. A trivial example game: 1. e4, a6 2. Bc4, a5 3. Qh5, a4 4. Qxf7, Kxf7 ... Black wins.
4. Why play this variant?