While Franklin Armory states their BFSIII AR-C1 Trigger is compatible with most BCG's, it is recommended to use a full-auto BCG as they reduce "bolt-bounce" as they are heavier. However, you can still use a non full-auto BCG and it will function as intended.
You do not need a full auto BCG for this trigger to function. As long as your BCG can operate a standard Milspec AR trigger it should function with our BFS trigger.
No, the trigger should function with any BCG that can properly function a Milspec AR trigger. The BFSIII can operate with a semi auto cut BCG.