Is a queen for two rooks reasonable in this position? What are the disadvantages? Is Qxe8 reasonable?
I always wanted to trade off a queen against two rooks and the engine likes this idea, but am I leaving my pawns unprotected in this situation? 
It would be hard to protect those with rooks.
This is a losing position for White. Normally, trading for a queen against two rooks is reasonable but Black has an extra knight here...

The best objective move sequence would be:

1.Qxe8 Rxe8 2.Rxe8+ Kf7 3.Re2 Qb6+


White's position after 3...Qb6+ is absolutely hopeless - the rooks have no useful squares. White has no counterplay, no attacks, no chance for Black to make a mistake. Very easy win for Black.

If this were a GM game, the White player would have resigned. Here, your best swindle move is 1.Qd3. By keeping the queen on the board, you --might-- be able to generate a miracle comeback.

A possible line:

1...Nxf4?? 2.Rxe8! Nxd3 3.Rexf8# checkmate!


The above line is not impossible in a blitz game. But it wouldn't have happened if you played 1.Qxe8.

Don't follow the computer analysis. Computer lines are meaningless in a lost position. You will need to try a human move. Don't trade your queen for the rooks. You need it to give troubles for the Black players.