Computing the intersection of two NPDA where it is possible
I think this is possible for the subclass of CFLs that are permutation-invariant with a binary alphabet. 

These correspond to type $\langle 1,1\rangle$ quantifier languages comparing the cardinalities of two sets. [1] characterizes such languages accepted by DPDA by the equivalent semilinear sets, and states at the end that quantifier languages accepted by NPDA are finite boolean combinations of such languages accepted by DPDA. 

A theorem of van Benthem ([2]) says that pushdown automata accept type $\langle 1,1\rangle$ quantifiers definable in Presburger arithmetic (i.e. defined by semilinear sets). So, if you get two languages that are non-deterministic CFLs (using the first paper to know you have such examples), their intersection should also be a CFL by this theorem.

The semilinear set which is their intersection might be a bit difficult to compute... but, if you have that, [3] (pgs.11-12) give an algorithm for creating an NPDA accepting the language based on the generators of the corresponding semilinear set. 

[1] Makoto Kanazawa. Monadic quantiers recognized by deterministic push-
down automata. In Proceedings of the 19th Amsterdam Colloquium, pages
139-146, 2013. 

[2] Johann van Benthem. Essays in Logical Semantics. Studies in Linguistics and Philosophy Volume 29, 1986, pp 151-176.

[3] Marcin Mostowski. Computational semantics for monadic quantiers. Journal of Applied Non-Classical Logics, 8(1-2):107-121, 1998.