The answer to question 1 is no, because the theory of the minimal transitive model itself is parameter-free definable as that theory, and in the other answer I explained why this theory is not an element of the minimal model. More generally, one cannot in general make much of a conclusion about an object from it being definable, becauase in the light of the universal definition, any object can in principle be made definable in a forcing extension.

The answer to the second question is yes. Because the minimal transitive model of ZFC is transitive, it has the true $\omega$ and thus it has the true ZFC and all finite extensions, and it agrees on consistency statements since it has all the same proofs that we do.

I don't understand the third question (or your remarks about subsets on the other post). I'm not sure about what kind of models you intend---do you intend submodels of the minimal model $\langle L_\alpha,\in\rangle$, that is, with the same $\in$ relation? All such models would be well-founded and thus would have transitive collapses either to a set in $L_\alpha$ or they would be $L_\alpha$ itself.