For (1) note that it isn't true forsquarematrices, so you have to think of non-square A and B.

For (2), note that over Z/2 an antisymmetric matrix is also symmetric and that the sum of symmetric matrices is symmetric.

For (3) write A^2 = (AB)(AB) = A(BA)B = ABB = (AB)B = AB = A.

For (4) consider the degrees.