If she is submissively peeing, then harsher corrections (prong collar) may not be the way to go with this puppy.
Also the fact that she will not take treats during training class -- this indicates the puppy is nervous moreso than aggressive/defiant/high energy/high prey drive.
What I would do with this dog, is I would go to the class, sit on a chair away from the class, far away from the class, and have something uber good, microwaved hotdogs or chicken or cheese, and offer the puppy a tid bit when it pays attention to me. As the dog becomes more relaxed further away, I would get a little bit closer.
Barking, lunging, pulling on the leash, distraction to the point of not being able to focus all sounds like a dog that is fearful and uncomfortable in the situation.
5 month old puppies are rarely defiant. They have untapped energy, they are intelligent but have short attention spans, they are curious and explore the world with their mouth and teeth. 1 mile twice a day is simply not enough exercise. You need to get out there and throw the ball through the snow and have that pup bound back and forth and back and forth. If he won't play with the ball, you play with him. run back and forth with him. Get some of that energy out before going to class.
If you treat a soft or handler-sensitive dog in the same way you would treat a hard, high energy, high drive, or defiant dog, it can backfire badly, especially for the dog. For a soft or handler-sensitive dog, you want to build their confidence in themselves and in you. What increasing corrections will do is it will make such a dog less confident, and more worried. You might be able to mask some of the symptoms you are seeing, but down below you will have a fearful dog who might react very poorly in common situations.
Bear Cub, Hepsi-Pepsi
Cujo2, Karma Chameleon
Ramona the Pest, Kojak -- who loves you baby?
Tiny Tinnie, Susie's Uzzi, Kaiah -- The Baby Monster.