Hope the little pup is doing OK and you were able to get her to the vet.
I would be surprised if your vet didn't want to start immunizations over from the beginning given the pup's background. I've heard horror stories about the vaccinations some of the "breeders" give, from the vaccine not being handled or stored right to using the same needle to inject multiple pups.
When you got the pup, were you given some of her food to take home with you? Blue is a good food, but if you suddenly switch foods pups and dogs can get stomach upsets. It is advised to mix the old food and new food over the course of a week or so (start with 75% old to 25% new, then 50%/50%, then 25% old to 75% new). The mucus on the stool could be related to a mild bout of colitis from the food switch. Or parasites, or other things. As you planned, getting this checked is advisable.
My pup was so little that when I fed him, he looked bloated as well. I ended up dividing his meal into 3 or 4 portions. Made housetraining a nightmare, but was better for his little tummy. And it is only for a few weeks. The lethargy and lack of interest in food is concerning, however.
I know several are questioning where you got the pup, plus your comment that the person was "so-so". I got my pup in September at age 9 weeks. He had been given two sets of immunizations and wormed appropriately. I was given a packet of information that included his health records. This had stickers from the vaccination vials, information about the wormer used, weekly weight listings and a printout from the vet detailing my pup's health and the overall health of the litter. I was also given heartworm preventative, a month's supply of Advantix (suitable to his weight) and a bottle of vitamins. This is in addition to all registration papers, etc. Just want you to know that you should not need to ask for proof of the vaccinations, this should have been given to you at the time of purchase. My vet typically will not accept breeder-given immunizations, however, he did in my pup's case. All the needed information was provided along with the breeder's contact information. My vet knew the breeder was a former vet tech and understood how to handle the serum.
Hope you will check back with us and give an update. Good luck.