Two rook problems I was thinking about chess problems and I couldn't find an answer for these. Are those open? If yes, how good upper or lower bounds are known? How many rooks one can put in a normal $8\times 8$ board such that every rook threatens exactly 2 other rooks? How many rooks one can put in a normal $8\times 8$ board such that every rook is threatened by odd number of other rooks?

I think it is good to ask both questions in a general way: How many rooks one can put in a normal $8×8$ board such that every rook threatens exactly $k$ other rooks?

If $k=1$ then the lower bound for the number of rooks is 2. But for the upper bound, we must find different pairs that mutually threaten each other. So we can locate 4 pairs or 8 rooks on the board.

If $k=2$ (your first question) then the lower bound is 4. I think the upper bound is 30, where we put a rook on the margin squares (a and h columns and 1,8 rows).

Now I prove that $k$ can not exceed $2$. Indeed if $k \ge 3$ then every rook must be in the middle of two other rooks in a row or column which is impossible (note that this must happen for all rooks in the board).