1. The APS theorem works for any  Dirac-type theorem; see  e.g. the excellent monograph  by Booss-Wojchiecowski on this topic.


2. More than four decades  ago, Boutet de Monvel has described a general set-up for dealing  with boundary value  problems thatmimicks  the K-theoretic approach to the  index theorem on closed manifolds.   For a  modern presentation  of this point of  view I recommend this  paper by  Melo-Shrohe-Schick  [arXiv: 1203.5649][1]   and the references  therein. It also involves noncommutative  geometry because  the symbols in the Boutet-de-Monvel calculus of boundary value  problems   belong to a more  complicated algebra than the symbols  of operators on compact manifolds.


  [1]: http://front.math.ucdavis.edu/1203.5649