Situation 1: Germany captures KAR turn 3.Â UK moves to SZ6 on turn 3, does UK roll for mines?
No. Mine fields are only active while the naval base is under the control of either its original controller or another power on the same side.
Situation 2: Germany captures KAR turn 3.Â If GE still controls turn 4, can they repair in SZ12 and mobilize new ships in SZ12?
Germany can repair battleships there, but not mobilize sea units. New sea units may only be mobilized at a naval base that the power originally controlled (not captured ones).
Situation 3: Germany captures KAR turn 3.Â Russia contests in turn 4.Â Can Russia mobilize ships in SZ12 (or SZ6)?Â ALSO, any mines go back to Russia, right?
Yes to both, assuming that Moscow is not controlled by the Central Powers.
In the FAQ (and rules) under Sea Zones, in the last sentence there is a phrase “(origional territories only)”, what is that trying to imply?
It means that the naval base must be located in a territory that your power originally controlled (at the start of the game) in order to be used for that purpose.