Firstly I'm sorry for your health struggles.
As far as the issues raised, I think I can see the issue your son has if you were asked not to come to the hospital yet turned up anyway and tried to get round the fact he had said no by waiting outside. If they wanted exclusive time with their new baby for the first couple of weeks or even months then that's ok and they have earned that right, it's their baby after all.
As far as DIL's mother being there, DIL had just given birth and was no doubt feeling vulnerable - it makes sense that she would want the woman there who raised her from being a baby. Her mother is probably the closest and most trusted woman she has in her life so, of course, she would naturally go to her while needing support or trying to navigate motherhood (in the same way your son may go to a male relative of his own before ever asking DIL's dad re advice on fatherhood). You have come into her life much later in life so the 2 relationships won't compare. She likely just wants her mother around more.
Irrespective of who puts pics of the baby on social media, it's still be best practice to ask the parents before posting their child - after all, it is their child and not yours.
As far as the dog, just because you choose to trust your dog around the baby, it doesn't mean your son has to (I personally have to exceptionally well trained working dogs and yet still exercise caution around my child as it's just the sensible thing to do)
Although it sounds like your son had a great childhood, it doesn't sound as though he's particularly close to you now but, as the old saying goes "A son is a son til he takes him a wife, a daughter is a daughter for all of her life" so he's just grown up and is advocating for his nuclear family.
I understand you have health issues but maybe should focus more on yourself and what makes you happy as opposed to putting all your energy into wanting to be 'Grandma' when it sounds like (at the moment) they don't want your involvement. The more you push, the more overbearing you will seem to them. The last thing you want is for your son to feel emotionally manipulated into giving you more access BECAUSE of your health struggles.
A baby of that age will only bond with parents initially anyway (they don't actually realise they are separate from their mother til about 8-9 months of age) so grandparent/grandchild bonding comes much later.
I would just give them the space they are clearly asking for and respect their wishes whilst focusing on your health and hobbies that make you happy.
Good luck with everything and I hope your treatment is a success.