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:

$$\require{AMScd}\begin{CD}
A  @>>>  B  @>>>  B/A  @>>>  \Sigma A \\
@VVV  @VVV  @VVV  @VVV \\
C @>>> D  @>>>  D/C  @>>>  \Sigma C \\
@VVV  @VVV  @VVV   @VVV \\
C/A @>>> D/B @>>>  X @>>> \Sigma(C/A) \\
@VVV  @VVV @VVV   @VVV \\
\Sigma A @>>> \Sigma B @>>> \Sigma(B/A) @>>> \Sigma^2 A \\
\end{CD}$$

The bottom right square anticommutes, but the rest commute.  The maps on the bottom and right edges have minus signs.