#1, flea treatment does not kill fleas immediately. They don't catch on fire when they hit the body of a treated dog, they can live for a few hours. The treatment will eventually kill them, as well as prevent them from laying eggs/kill eggs.
#2, the vet HAS to give the dog to the rescue if they ask for it, they still owned the dog.
After speaking with someone who finally knows just a little about this rescue, the people who run it are kind of wussies. Most likely they denied the app, but were passive aggressive when getting the dog back. The vet has to call the rescue for payment and when they called the rescue probably said "oh we will come get the dog"