As Black, how should one respond to 4. Qe2 by White in the Russian Game, Damiano Variation? I recently played a game as Black. My opponent and I transposed into, according to Lichess, the Russian Game, Damiano Variation, when White played 4. Qe2. I wasn’t sure how to reply, and I ended up losing my queen and soon resigned.
[FEN &quot;&quot;]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4 4. Qe2 Nf6 5. Nc6+ Be7 6. Nxd8 Kxd8

As Black, how should one respond to 4. Qe2 by White in the Russian Game, Damiano Variation?
Do not play 3...Nxe4 in response to 3. Nxe5.

I feel this important aspect is missing in the other answers. You can see the primary reason it doesn't work in your game - you end up losing the queen. Given that you can't just move the knight after 4. Qe2, your next-best option is to play 4...Qe7 5. Qxe4 d6 (or ...f6), which still results in you losing a pawn. It's not a disaster because Black does have some compensation in the form of superior development, but Black's position is still undoubtedly inferior.

Avoiding this line is the main line of the Petroff, and is much more stable. The main line goes 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5.d4 d5. Black has gained a tempo (as seen from his advanced knight), but the knight is somewhat exposed and can potentially be forced to retreat, in which case White is up two tempi. In practice, White can usually achieve this, but needs to play pawn breaks like c4, which balances out. The Petroff is thought to be drawish for a reason.