Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary? I recently learned about the Classification Theorem for compact 2-manifolds.  Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?
Moreover, is there a theorem which classifies the 2-manifolds with boundary?

Yes, there's a classification theorem for non-compact 2-manifolds.  
This paper gives the classification for triangulable 2-manifolds:
http://www.jstor.org/stable/1993768
That an arbitrary (2nd countable, Hausdorff) topological 2-manifold admits a triangulation is fairly classical.  Ahlfors book "Riemann Surfaces" has a proof.  There are others available, see for example this list:
https://mathoverflow.net/questions/17578/triangulating-surfaces
If all you're interested in is compact manifolds with boundary, you get that classification immediately from the closed manifold case.  Because if you have a compact manifold with boundary, its boundary is a disjoint union of circles.  So cap those circles off with discs to produce a closed manifold.  So compact manifolds with boundary are classified by the closed manifold you get by "capping off" and the number of boundary circles you started with. 
Non-compact manifolds have a more delicate classification -- think for example about the complement of a Cantor set in a compact surface. 
