If a person cuts back or stops eating fat & oil, does the gallbladder simply stop functioning or barely functioning?

It will reduce gallbladder function.  But there are other factors that also affect gallbladder function such as bitters and hormones.

Does the gallbladder squirt bile to digest all fat & oil eaten? I ask because I've read that coconut oil is processed differently in the body than other fat/oil. If coconut oil was the only fat I was eating, would my gallbladder still continue at full function or would it cause my gallbladder troubles?

As far as bile release goes it has been shown that coconut oil has the same effect on bile secretion as other oils.  Although, coconut oil is a natural surfactant, which does change its absorption properties compared to other oils making it easier to absorb, even in the absence of bile.