Classification Theorem for Non-Compact 2-Manifolds? 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.

Here's a summary of the situation regarding noncompact $2$-manifolds with boundary, thanks to Moishe Kohan and Jacques Darné.

Originally, I posted an answer pointing to the 2007 paper Classification of noncompact surfaces with boundary by A. O. Prishlyak and K. I. Mischenko, Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66. However, Kohan pointed out that the classification had actually been completed much earlier by E. Brown and R. Messer (The classification of two-dimensional manifolds, Trans. Amer. Math. Soc. 255 (1979), 377–402). Then more recently Darné pointed out that the theorem claimed by Prishlyak and Mischenko is false, because it contradicts the one of Brown and Messer. See Darné's comment below for details.

So the upshot is that the correct classification of noncompact $2$-manifolds with boundary was completed in 1979 by Brown and Messer in the paper cited above.