Can we interchange columns in a determinant like this and preserve the value of it? For example: $$ \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 1 & 0 \\ 3 & 1 & 1 & 0 & 0 \\ 4 & 1 & 0 & 0 & 0 \\ 5 & 0 & 0 & 0 & 0 \end{pmatrix} $$
to $$ \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 2 \\ 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 5 \end{pmatrix} \cdot (-1)^2 $$
Yes, we are allowed to interchange the columns of a matrix, but we must introduce a minus sign upon each swapping of a pair of columns. So, your calculation above is correct. The rules of row operations are the same as the rules of column operations because, as @Paul says in the comments above, $\det(A) = \det(A^T)$.