I do agree about the vaccinations. Bordetella and leptospirosis should not have been given so early in my opinion, normally they should be given at approx 12 weeks (second set of vaccinations) I am not familiar with them being given earlier. Normally puppies are given 3 sets of vaccines (8, 12 and 16 weeks). It was a lot for him to get in one bundle for his age and size. However, for a shelter pup, I can see why bordetella may have been given earlier because kennel cough can easily be aquired during the stay at the shelter.
As for probiotics? You've gotten some good suggestions. Adding a probiotic to the diet each day is not going to be harmful - it's like us eating yogurt every day. There is also Forti-flora, a veterinary product, its a probiotic powder that is sprinkled over the food.
Puppies commonly have parasites. He may have not been dewormed or not properly dewormed at the shelter. If he is having soft stools, you can choose to deworm or take in a stool sample and have it analyzed (and treat according to any findings) and I would continue to have a stool sample checked routinely (with each visit take in a fresh stool sample). Any parasite eggs will be identified on a fecal flotation. I recommend taking in a stool sample to your veterinarian to rule out parasites.