Question: Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of short exact sequence. Can we use this to define ext groups which classify extensions? What works and what doesn't work and why?

Answer: 
I can't give a substantive answer to this question, but differential graded abelian monoids have been sighted in the wild here:

A Chain Functor For Bordism 
Author(s): Stanley O. Kochman 
Source: Transactions of the American Mathematical Society, Vol. 239 (May, 1978), pp. 167-196 
Published by: American Mathematical Society 
Stable URL: <http://www.jstor.org/stable/1997852>
Presummably you could use the forgetful functor to sets and its adjoint to construct a homology theory (using the walking adjunction, simplicial sets, etc) analogously to how group homology arises from the forgetful-free adjunction to set.

The details are probably rather dense and tedious though.