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:
$\begin{smallmatrix}{ccccccc} A & \to & B & \to & B/A & \to & \Sigma A A$
$\downarrow \qquad \downarrow & & \downarrow & & qquad \downarrow & & \downarrow quad \qquad \downarrow$
$C & \to & D & \to & D/C & \to & \Sigma C C$
$\downarrow \quad \downarrow & & qquad \downarrow & & \downarrow & & quad \qquad \downarrow \qquad \downarrow$
$C/A & \to & D/B & \to & X & \to & \Sigma(C/A) Sigma(C/A)$
$\downarrow \quad \downarrow & & qquad \downarrow & & \downarrow & & quad\qquad \downarrow \qquad \Sigma qquad \downarrow$
$\Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A \end{smallmatrix}$A$
The bottom right square anticommutes, but the rest commute. The maps on the bottom and right edges have minus signs.
