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 Faisceaux Pervers and possibly also the triangulated category chapter in Weibel's Homological Algebra.
Edit: I think there is a way to write a Mayer-Vietoris sequence as chains on the totalization of a big grid like this. Then you need the signs to make it work. Here is an attempt at a diagram:
$A \to B \to B/A \to \Sigma A$
$\downarrow \qquad \downarrow \qquad \downarrow \quad \qquad \downarrow$
$C \to D \to D/C \to \Sigma C$
$\downarrow \quad \qquad \downarrow \quad \qquad \downarrow \qquad \downarrow$
$C/A \to D/B \to X \to \Sigma(C/A)$
$\downarrow \quad \qquad \downarrow \quad\qquad \downarrow \qquad \qquad \downarrow$
$\Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A$
The bottom right square anticommutes, but the rest commute. The maps on the bottom and right edges have minus signs.