Welcome to YT
The coughing or choking sounds like kennel cough or collapsing trachea.
The itchy leg and butt could be so many things. Fleas, mites, worms (for the butt atleast, alleriges...
The runny stool could be stressed related but I wouldn't accept your vet's advice to just watch. It can be very serious in a puppy. I would go to a new vet and switch foods (very slowly because this could aggravate it more). My first choice is to homecook.
Because you pup was a rescue, please get him bile acid tested.