Is there any variant where both players play blind every other move or in the starting?

There's a 'fog of war' variant where you can only see the squares that a piece of yours could move to.

That one would be interesting to try with AlphaZero. Reinforcement learning systems tend to have trouble with imperfect information.

--that a piece of yours could attack, I believe, not move to

I just checked and if you want to be technical it's "move to normally, could move to during a capture if a piece was there, or holds a piece you could capture".

Pawns are the only case that where those three are different but they see all three:

- directly in front (move)

- forward-diagonal (normal attack)

- horizonal exactly only in the case of a enemy pawn moved to that square last turn that can be taken en-passant (capture). That square goes dark if you move another piece.

pychess.org has many chess variants, and you can play against the AI to try them out. No registration required.