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: 
Your question can be understood as how to do Homological Algebra over the Field with one Element.

Deitmar, in <http://arxiv.org/abs/math/0608179> , section 6, gives an example of what can go wrong if you try to do sheaf cohomology directly via resolutions...

You might also want to look at his <http://arxiv.org/abs/math/0605429> ; in order to construct K-theory of monoids he sets up an analogue of the Q-construction. The Hom-sets in the resulting category are sort of Exts, maybe something to start with...

Durov, in <http://arxiv.org/abs/0704.2030> , follows the simplicial approach for commutative monads, of which commutative monoids are a special case
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).