Correctness in Multi-party Computation: Real/Ideal Paradigm in Malicious case
1- $\:$ I believe correctness does not have an exact meaning in this context.

It would certainly involve not letting the distinguisher see the transcript,

and might involve not letting the distinguisher see the adversary's randomness,

and possibly involved not even letting the distinguisher see the auxiliary input.

2- $\:$ [Computational/Statistical/Perfect] correctness is implied by

[computational/statistical/perfect] indistinguishability of the Real and Ideal cases,

since the distinguisher gets to see the honest parties' outputs.