HOWEVER, what happens during next production phase ? Obviously there is no battle (not until the movement phase), but do you keep control of the buildings (and therefore their production ?
The province will be At War, as more than one player will have their units in it. The buildings will produce for the player who has Control Marker in the province.
Similarly, could you try to convert a region where you Petrify an enemy, if you managed to kill the (back to normal) units on the NEXT turn ?
Only if their owner goes before you in the Battle Phase and then he either exits the province or tries to kill your units and fails but will get all his own perished (this is also valid for any other players' troops that may be present in the province). If you go first, you won't be able to take over the Control Marker, as the province convertion condition (only your troops present, one of them on the Control Marker) is being checked precisely at the start of _your_ Battle Phase.