One condition equivalent to integrability of the almost complex structure is $\partial^2 = \bar\partial^2 = 0$. If we're just talking about quantum mechanics with a Kähler target, then the Hilbert space is the space of complex-valued differential forms on the target with integration against the symplectic volume form giving the Hilbert space pairing. Then some combination of the supercharges act as $\partial$ and some as $\bar\partial$. The $N=2$ algebra relations imply the integrability condition above.

I think that for a symplectic target, while it is possible to define the A-model, it is not a topological twist of a well-defined $N=2$ sigma model.