Mom can't take care of the dog, any guidance?

For her doctor to place the burden on you of saying she should live with you is outrageous.
We can agree that mom should not be living alone.
The options then are..
*caregivers hired (paid for by mom) and care for her 24/7.
*placing her in a Memory Care facility
*moving in with a family member that has a house that is SAFE and will accommodate her as she declines AND is willing to care for her. And caregivers will be needed to help out as well.
There is no easy answer, any of these would be a tough decision. Caring for someone with dementia is not easy. You must think about your life, your family.

Your mother needs the evaluation and then a solid talk with her doctor. She needs to be told (with confirmation in writing) how she did in the evaluation and what the doctor says about her belief that she’s ‘doing well’. She may not take it in or remember it (the written confirmation may help), but she deserves to know. And you deserve to know that whatever she thinks or remembers, she has been told all the true information and it isn’t just you being nasty. If you have family or others who will blame you for ‘putting her away’, give them a copy of the note.

I agree it is now dangerous for mom to live in her home alone. You should send APS (adult protective services) to check on her. For the short term someone should be hired (paid for by mom) to come in and care for her and the dog.

Mom also needs to be evaluated for what she needs (check with the Area Council of Aging or Social Services in her area. For long term, you and the other family member need to move mom to the appropriate care facility. Research on line and narrow down the top contenders and take some time to visit them to find the facility that fit her needs the best.

The other option is to move mom closer to you or the other family member. Each of you should also look at your area for care facilities.

This is a lot to take in and do in a short amount of time. Take a deep breath and work on it one step at a time. Best of luck.

Dare I ask...can this neighbor be enlisted to help with this pup? Do not underestimate the support and love this pup offers in the absence of others and the impact the loss would be on your mom even if you were able to take the pup. Also wondering if this might be a meddling neighbor tending to the dramatic and if the pup just spit up a little which they sometimes do if they drink too fast or too much. Neighbor might think mom belongs in assisted living and generating a story to fit...maybe she has a friend or family member that would like to live close by/next door??? YOu never know. And so let's say mom does go, into assisted living, and you have separated her from her beloved pup, and then what? You reduced her being responsible for another life, but she is still going to be prone to memory loss issues and all the rest. Assisted living is not the answer all the time. Perhaps hire a pet sitter who can discreetly check on mom as well with 1-2 daily visits.Maybe a meal brought in or going out once in a while? Also they have some dog stuff that looks like a plot of grass...and pups can be trained to use it where placed...inside or on a balcony etc. How has mom managed previously with lousy weather and walking pup??

Obviously if your mother has a short term memory that last 10 seconds she needs to be placed in memory care not assisted living. Assisted living is not for seniors with dementia. The dog should be taken to get and not brought back to your mother. Will you or other sibling be willing to take the dog in? The animals always seem to suffer when seniors decline because no one wants to upset said senior.

Yes, its time to place Mom. If she can't care for an animal, she is not caring for herself.

I would say that would be past the last straw. She is a danger to herself and the poor dog if her memory span is 10 seconds. She can turn on the stove walk away and forget, etc.
It is in her best interests to be somewhere that she can be looked after and I hope the same goes for her dog. In the meantime, is there someone who can take the dog to the vet?

Ask the neighbor to get the dog to the vet, and then rehome him/her.

Time for placement for her.