If you take a commuting square of maps and use homotopy cofibers to extend outward to a big grid, you eventually run into some anticommuting squares, which are really supercommuting squares.  I think this answers your last question in the affirmative.  Whether this answers your other questions really depends on your standards, I suppose.

The anticommuting square is mentioned in <i>Faisceaux Pervers</i> and possibly also the triangulated category chapter in Weibel's <i>Homological Algebra</i>.