I'm sure there are lots of situations you've experienced that you haven't listed here, but the one that you did doesn't explain the situation very well. There is no reason your parents, who live 4 hours away, should have to give up their pets. To ask them to is quite unreasonable and illogical - it is not like you need her to babysit or that she is your nanny, and your son spends weekly visits there - she lives a long way away and those dogs are her companions. It seems pretty selfish of you to ask her to give them up so that you can occasionally visit without staying in a hotel.
However, having said that, she needs to understand the reality of your situation. Putting the blame on your son for his allergies is not helpful or useful, and while some people do outgrow allergies, people also grow into allergies (I was diagnosed with my dog allergy in college, after having dogs all my life. Luckily, it is manageable, while your son's seems to be at least currently very severe). There is no guarantee that you son will outgrow this allergy, and you need to make that very clear to her. Any time she asks, repeat, "He won't outgrow it, Mom. As long as your dogs are alive, he won't be able to come stay with you. I'm sorry, but you can always visit us here." Don't be angry or frustrated with her, or say she chose her dogs over your son - she didn't. But certain sacrifices come with having them.
As for her health - well, sadly, you can't do anything about that, except perhaps be a little bit less sympathetic. When your mom begins complaining, tell her you're sorry that she is unwell, and perhaps a change of diet or walking the dogs more regularly would help. Then tell her you have something you need to do, it was good to talk to her, and hang up. Repeat as many times as necessary. If she doesn't get the sympathy she wants from you, she should eventually get a clue. If not, well, at least you won't have to listen to it.
I really liked someone's idea about using a webcam. Then, your mom can see and talk to your baby more often. But living 4 hours apart and having your own family, it is only natural that you and your mother will grow apart some. Keep trying to keep that connection, but don't put up with behavior that you find offensive. Since you are so far apart, when things start to get uncomfortable, hang up. It's hard for us (we always feel like we have to take care of our parents), but your mom is a big girl who managed to raise you, and she'll have to take care of herself now, too. Good luck.