Intersection of two NPDAs
The intersection of two context-free languages can be non-context-free. The classical example is
$$ \{ a^n b^n c^m : n,m \geq 0 \} \cap \{ a^m b^n c^n : n,m \geq 0 \} = \{ a^n b^n c^n : n \geq 0 \}. $$
So in general you cannot simulate the intersection of two NPDAs with an NPDA.