The dad was a retired fighter pilot that had late-life sex-reassignment surgery. The A-plot of the episode was Quagmire unable to come to terms with the fact that his hyper-macho dad was a transsexual woman.

Brian is treated inconsistently based on the comedy needs of the episode. Sometimes he's treated like a dog, but in recent seasons they've been treating him like a human most of the time, with the occasional surreal gag that points out the fact that he's a dog. But he drinks martinis, has human girlfriends, drives a car, wrote a crappy pretentious novel that got panned by the New Yorker, etc. Basically anytime they need a straight man for a romance subplot, or a male character that isn't a buffoon like Chris or Peter, they use Brian the dog.

Within the fiction of the show, it's not at all unusual for the dog to be having sex with a human woman.