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: 
This is an answer to one part of your question. The paper “Extension Theories for Monoids” by Charles Wells, Semigroup Forum 16 (1978), 13-35, gives a precise answer to the specific question: How does the Beck cohomology theory for monoids classify extensions of monoids? (It classifies Leech extensions.) The paper with corrections and a list of subsequent papers related to it may be found [here](http://www.cwru.edu/artsci/math/wells/pub/pdf/ExtThMon.pdf). Beck's thesis is now online [here](http://www.tac.mta.ca/tac/reprints/articles/2/tr2.pdf).
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>