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>.

<b>Edit:</b> 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.