Is my dog being a bully or just a puppy?

We adopted a 7 week old small mix puppy from a shelter a little over two months ago (he is now 4 months old.) He has always been a biter – he loves to gnaw on our hands, and will also bite our pants and drag behind us. We have tried sharply saying, “No!”, spraying him with water, holding his snout shut while looking in his eyes and saying “NO!”, flipping him on his back and holding him down until he calms down, etc. For the most part it seems playful, but as he gets older it is starting to really hurt sometimes.

He has played with a few friends’ dogs and likes the dog park, and behaves pretty well there, but all the dogs are bigger than him. When he plays with his “cousin,” a french bulldog, they both sometimes growl at each other and bark but haven’t really fought. My dog kept stealing his cousin’s bone and running away though.

And now, we are dogsitting a 3 month puppy for some friends (also a small breed.) They seem to love wrestling each other, and the other dog holds her own against our dog, but twice they have started sort of snarling and growling and seeming like they’re really biting each other. My dog also barked and nipped at the 3 month old when she started eating from his food bowl (she had her own.) What really worries me is that when they’re playing and biting at each other, my dog’s face looks like it is snarled up and his lips are pulled away from his teeth. He has also done that to me and bitten at my hands when I pick him up when they’re in the middle of wrestling.

Is this just puppy stuff? He is so small it is still manageable but it seems like this is what leads to “small dog syndrome.” I want him to be very friendly and well-behaved, and I’m worried this is a bad sign… am I over-reacting or should we be taking drastic behavioural measures? In general, what can we be doing to curb his biting and make him play nicely with other dogs?

(To his credit, my dog is generally a very loving and happy dog, loves people, loves licking faces, sleeps under the covers with us at night, is making good progress on potty training, knows several commands already, wags his tail incessantly when we come home, etc.)