Can you make up any matrix permutation with switching rows, columns, and transposition? If I have a matrix of an arbitrary size, can I transform it into any matrix of same size that uses same elements using only switching rows/columns, and transposition (or transposition but using the other diagonal)? I looked into this answer, but it only says that it's impossible without transposing the matrix.

It's still impossible to go from, for instance, $\left[\begin{smallmatrix}1&1\\0&0\end{smallmatrix}\right]$ to $\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$, because your operations don't change whether your matrix is invertible. In fact, they don't change the determinant at all except perhaps a sign change.
More concretely, if two element are on the same row or column, no matter what combination of operations you do, they still share either row or column afterwards.
