Yes, play biting is one thing (which I think you are describing in part) but biting when being picked up or when being told no or being bothered is a different kettle of fish and something you want to take steps to diminish. It is understandable to a degree that a dog may not like certain things but the hope is that they come to tolerate this at times graciously rather than resorting to using teeth.
For the time being however you probably want to avoid provoking those situations so that your dog doesn't keep practising this behaviour, and so that you can put a plan of action together about how you are going to train and manage these situations. So teach your son to avoid behaviours that might put him in danger (I don't think I saw how old he is?). If you feel it can be safely done I would be involving your son in feeding and training - if he is the one giving food to the dog (and issuing commands first - sit, lie down etc) the dog may come to respect him a bit more. Similarly with training for treats - it will help him establish some authority in their relationship dynamic.
I would try to instill this as part of the household routine - the dog has to do something to get something they want, be it, food, out for a walk, play time etc. 'Nothing in life is free' - google it, a sort of dog training technique, but it will help you establish a better balance in the house,as at the moment the dog is telling you what she finds to be acceptable behaviour, not the other way around.
We had all of your problems with our dog, especially at the age yours is now, but he is so much better now at 18 months (and was well on his way to this at 12 months).
When you feel ready you can approach some of the situations that currently cause the dog to react in a new way to help train a new response. When Ollie was snapping/ biting at being lifted as a pup we started a training regime with my partner and I both being present - one lifting in a new way (sideways on, arm under rump and chest) which is more comfortable for dogs, whilst the other occupied the head area with treats. The treats should be good enough that the dog is more interested and focussed on them than she is bothered about being picked up. We did this 2 or 3 times a day for a couple of weeks and then maybe once or twice a week after that so he did not relapse to his former snapping response. During training we would simply avoid lifting him until we could get all these conditions right - the 2 of us had to be present, the treats had to be good enough, because we did not want to risk a bad lift that would set him back again. Now he can be lifted by one or other of us with no bother at all.. we still give him treats and make a fuss when we do.
