If your daughter is not a resident of your household but is named on your insurance how involved ar you if she drives her boyfriend's car?

I'm not sure I understand the question.

If her boyfriend has insurance, then she is covered under his policy as long as she had permission to drive his car.

As far as your liability if she has an accident, it all depends on the extent of the accident and if a law auit is filed for more than her boyfriend's policy limit. Then there could be potential to attack your insurance policy which names her as a driver.