When Barbara first changes into Batgirl, her cowl has a chinstrap, but no chinstrap is evident when she's in costume.
Penguin plans to become immune to the law by becoming Commissioner Gordon's son-in-law. However, the police commissioner has no power to determine who will or will not be prosecuted, a matter in the purview of the district attorney. Indeed, if anything, the police commissioner's son-in-law would be under increased scrutiny.
Penguin kidnaps a minister (actually Alfred), intending to force him to officiate the wedding. However, in order for a marriage to be valid, the parties must first obtain a marriage license from the city government before the ceremony can take place.
After Penguin tells Barbara Gordon she will marry him or she has another option. Screen widens to show a life-size cutout of Commissioner Gordon at which time Penguin shoots the cutout several times. Unless all the apartments are completely sound-proofed, Commissioner Gordon, Chief O'Hara, Batman and Robin would have heard the shots because Penguin and party are in the next apartment.
Alfred poses as the minister when Penguin's henchmen kidnap him. Yet, despite having crossed paths with him numerous times both at stately Wayne Manor and other locales, Penguin doesn't recognize him.
Barbara Gordon is employed as a librarian, yet, she has a fairly impressive multi-room apartment, complete with rotating wall, secret hideout, and access to a hidden motorcycle freight elevator.