Siblings want to take mom's dog away from her. What do I do?

Do everything you can to help Mom keep her dog even if it means keeping it yourself and taking it for long visits. You are correct Mom will be devastated.
As an aside how much were the three of you helping your stepfather during this time. Not trying to be mean - just an observation.

I am not sure if you are meaning that the dog does not like most men. A shelter would really not be a good place for the dog, since it isn't adoptable. Even if the shelter is no-kill, your mother's dog will likely live out its life in a small pen. That wouldn't be fair if other options are available.

Are your brothers concerned the dog will bite someone? I wondered what their concern is about, since your mother is still competent. BTW, they cannot remove the dog without her permission if she is competent unless they can show the dog is a threat or is in danger itself. Does the dog stay indoors? Being 1.5 hours away is pretty far in dog-time. The person who is there would be the only one who could really care for it.

An hour and a half isn't too terribly far from your mother, but would be cumbersome to make every single day. I understand the pain she is in, and her limitations. Is she a candidate for knee replacement surgery? Has that option even been explored? As far as the dog goes, when a man is at the house, can the dog be penned up in a bedroom or utility room for a while? It does seem extreme to totally remove him just because occasionally there will be a man at the house. If your mother is capable of attending to his needs then it would be heartbreaking for her to lose her husband and then her dog too. You could take care of the vet visits as they come up.

Are you thinking you might move in with your mother, or move her in with you? There are lots of pros and cons to those actions. How is your Mother/Daughter relationship right now? When you become her caregiver, the dynamics change. You will eventually be all up in her "business" and she may resent that. Financially and hygienically speaking. You may find yourself wanting to take over the Mother role; how do you think she'll react to that? Just some thoughts to ponder.

You only asked about the dog. What does Mom want to do about the dog? Sibs have no say-so really. Everyone is entitled to their opinion, but only Mom's counts.

Peachy, we need more information. Since your mother needed someone to be her caregiver, what are her medical issues? Who will now be your Mom's caregiver? Could the caregiver help with care of the dog? Walk the dog if need be? Or will your Mom be moving to a higher level of care facility?

I can understand that it would devastate your Mom if the dog was taken away, too bad a family member couldn't have the dog at their home and the dog could come to visit Mom.

From Peachy, the only medical issue is bone on bone knees/arthritis. No other issues known of. I want to try to be her caregiver from now on and yes, the dog is no problem. The dog has plenty of exercise in woods next to house. I offered to help at any time with my Mother, he would not let me know how overwhelmed he was. I helped with Mom when he was in hospital and he was very appreciative. Other siblings live out of state and were unable to help with Mom. I lived the closest to them 1 1/2 hour away.

My Mom has Dementia and is not capable of taking care of herself or be left alone most of the time. I make her meals, bathe her and feed her dog. Also, she has not been diagnosed by a physician. We already know and will not put her through all the rigorous and costly tests. My sister and step-sister are executors of the estate at the present time. So not sure I will have any say about getting rid of Mom's beloved dog (Molly).