Hi, about dog foods, it sounds like your vet is trying to push the brands that he carries. There is an excellent website called Dog Food Advisor that has independent reviews and recommendations for most dog food brands, and Hill's Science Diet for Puppies gets only 2 1/2 stars (not recommended) out of 5 stars. Blue Buffalo Basics for small breed puppies and Royal Canin for Yorkshire Terrier puppies get 3.5 and 3.0 stars (recommended) respectively, and that is what I feed my Bella. Bella prefers Royal Canin because of its smaller kibble size, but unfortunately, I have a bag of Blue Buffalo (larger kibbles) she needs to finish up, so I mix the two foods.
Here is the website for Dog Food Advisor:
Dog Food Reviews by Brand
About brushing, my vet recommended that I wait until the adult teeth come in before I start thinking about that, so I've been putting it off
. If the vet saw red gums, they should have made a recommendation about what to do about that. If they didn't, call them back and ask.
For nail clipping, our vet did the clipping in about 30 seconds or less using the guillotine-type clippers. I was told that if your dog gets a lot of outdoor walking on concrete, that the nails will get worn down sufficiently and that nail clipping might not be needed. So far, so good, but I have a set of guillotine clippers ready for just in case.