1. The APS theorem works for any  Dirac-type operator; 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 that mimicks  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  involves some noncommutative  geometry because  the symbols in the Boutet-de-Monvel calculus of elliptic boundary value  problems define elements in the $K$-theory of a noncommutative    $C^*$-algebra. In the case of closed manifolds symbols  of elliptic operators lead to elements in the $K$-theory of a *commutative* $C^*$-algebra. 

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