Does anyone play this sometimes as Black?: 1. e4 d5 2. exd5 e6 3. dxe6 Bxe6


no, I've never played it and I think I never will.
why? well, because it's a gambit that doesn't give me a strong center and the bishop on e6 seems to be in contrast with one of the opening principles I follow most (knights before bishops)
the only gambits I play are Queen's Gambit and the Smith Morra Gambit. I don't even like the King's gambit, it openes the kingside too much.

It's a transposition to the Boehnke gambit.
I think it's a bit unsound. But if you want to play it play it with 2...e5
IM David Smerdon plays the Icelandic gambit. It was also suggested in "gambit opening repertoire for black" by Eric Schiller.

The original question poses something similar to icelandic gambit, but without the advantageous part. Here is how it is done (there are 2 ways to do it).
Note that defending with the knight allows the queen to come out without the strong tempo move and forces a weaker tempo move.
srn347 I believe that your first example is actually a transposition into the Caro Kann Panov-Botvinnik variation.
Clearing away BOTH center pawns should always be weighed very heavily. I really don't see the advantage of doing so.
Maybe I'd play this in a 5 min game, but in a serious game I'd steer clear.

In your variation (1.e4 d5 2.exd5 e6? 3.dxe6 Bxe6), white hasn't played c2-c4 yet, so this whole black plan is moot. White can always play something like c3 and d4, if necessary. Black is just a tiny bit ahead of development, which isn't close to enough for the gambited center pawn.