Can such a matrix exist?
No such matrix exists.

The Desnanot-Jacobi identity says that for $i \neq j$, 
$$
\det M_{ij}^{ij} \det M = \det M_i^i \det M_j^j -\det M_i^j \det M_j^i
$$
so using this, we get
$$
\det M_{12}^{12} \det M = \det M_{1}^{1} \det M_{2}^{2} - \det M_{1}^{2} \det M_{2}^{1}
$$
But your requirements force the left-hand-side to be 0 (mod 2) and the right-hand-side to be 1 (mod 2), showing they are incompatible.