Do different variants of Mergesort have different runtime?
Note first that we are talking about an algorithmic idea here; in order to apply a meaningful analysis, we would need at least pseudocode for the three variants.

Note furthermore that the given recurrence is most likely only valid in the limit. It is also not clear what the complexity measure is. Comparisons? In that case, it is clear that none of the adaptions change the measure at all. Basic operations? That would make more sense.


I agree with your reasoning; this seems to be the basic variant. Note again, however, that you'd have to look in the actual algorithm for its analysis. In particular, applying Mergesort in-place is uncommon.
Copying the entire array once adds a linear term globally, so the recurrence does not change. If you are only after $\Theta$-behaviour, ignore it (since the actual algorithm dominates).
Now this adds linear overhead to every call of the routine, changing the recurrence from $\ldots + n$ to something like $\ldots + 2n$. Of course, this results only in a different constant so the result is again not influenced (see Master Theorem).