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:

$\begin{smallmatrix}{ccccccc}
A & \to & B & \to & B/A & \to & \Sigma A \\
\downarrow & & \downarrow & & \downarrow & & \downarrow \\
C & \to & D & \to & D/C & \to & \Sigma C \\
\downarrow & & \downarrow & & \downarrow & & \downarrow \\
C/A & \to & D/B & \to & X & \to & \Sigma(C/A) \\
\downarrow & & \downarrow & & \downarrow & & \downarrow \\
\Sigma A \to \Sigma B \to \Sigma(B/A) \to \Sigma^2 A
\end{smallmatrix}$