Sorry, no puppies in this game. That happened in Back to Nature, which is the game on which Friends of Mineral Town is based, but they didn't carry it over to this game. In that game you could mate your dog with Barley's dog, Hannah, and then pick to whom to give a puppy. I gave my puppy to Gotz. I kinda wish they had put that in the game - it was cool.
Yup, unlike BTN, the puppy event with Hannah and the Inn event with the cat do not happen in FOMT. You get different prizes at the horse races though, and you can never miss a girl's heart event in FOMT if you're going for her, because you can only propose to her if you've seen them all and they happen even if her heart color is all the way to red.
o.O to bad you can´t ahve puppies, and in hm64 your dog had the puppies with the guy at the potion shop but he gives one of the puppies to a friend and keeps the other one o.O