Positivity and Interchange of Summation

Yes. Because either way it is equal to $\displaystyle{\sup\limits_{M,N}\;\sum_{n=1}^N\sum_{m=1}^M a_{m,n}}$.

It's worth knowing that rearrangements can change the value of sums or integrals only if the positive and negative parts both diverge to infinity.

Fubini's theorem says rearrangements are fine if both parts are finite.

Tonelli's theorem says rearrangements are fine if what's being summed or integrated is everywhere non-negative. (It follows that it also works if it's everywhere non-positive, since the minus sign pulls out.)

Putting the two together gives you what I said in the first paragraph.