I'm not sure I believe in proper classes and in particular, I'm not sure there's a proper class of all sets that could be the model of a second-order theory such that you could not describe any set larger than the model, and as for pinning down that model using axioms I'm pretty sure you shouldn't be able to do that. There are analogues of the Lowenheim-Skolem theorem for sufficiently large infinities in second-order logic, I seem to recall reading.