What should I play against the Zaitsev in the ruy lopez?

 10.d4 Re8 11.Nbd2 Bf8  and after you have a fairly tactical and sharp position.

You have the classical knight manoeuvre after nbd2, nf1, ng3 or e3 and if you want nf5.

You can have some sort of King side attack like Re3 and idea of a rook lift. Nh2 and after Ng4 and try to put your queen in the attack.

This is not the right question.

does this ever happen?

Yes, in the closed Ruy lopez, this is the 2nd variation that I face the most regulary.

After 10. d4 Re8 11.Nbd2 Bf8 you could also play 12.d5  Black will have some problems getting the c6 Knight back in the game.  Black will probably seek to undermine  your center with the ... c7-c6 break.  This break would reopen the a2-g8 diagonal for your White square bishop, and d5 could become an outpost for the your knight.  When the game breaks open you should seek to attack the Black kingside.