I'm taking care of my mom with dementia. Any advice?

Let's start with putting your worries about your little poodle to rest. They honestly are not worried at all by whom pees or poops where. If you notice how dogs greet one another, it is basically nose to butt. They don't suffer from our inhibitions and rules.
Now on to Mom. You don't tell us a whole lot here about your situation, other than to say you are going home, as you have been doing this for 9 months and are worn out. How long have you been giving up your OWN life to move in with your Mom and care for her? Has your mother always lived in the manner she lives in, or has this been a big change in her mentation that family members are now needing to address by stepping in one or another to stay with her?
Glad you have gone home with your own family and honestly it is where you belong. The real question is now whether your mother can continue to live alone, or whether someone must be with her full time? If someone must be with her full time does the family have any ideas about who might wish to do that? Is the family discussing together what should be done, perhaps considering placement?
And most of all, does Mom have dementia, or is she just not a real good housekeeper?
I hope you will come back, Sheila, and give us more information about your Mom, yourself, your family, and the current situation. We might have more ideas if we have more information.
Meanwhile, rest up. Hope you will give us more.

Your mom has dementia.
She can not be responsible for caring for her dog. She can not be responsible for caring for herself let alone a dog.
You can not change your mom the only thing you can do to make things easier for you is to change your thinking about what you expect of your mom.
Part of being a caregiver is to know when you need a break. You can get someone in to help you out. Not just for a weekend but for several hours several days a week. The other option is to consider placing your mom in a Memory Care facility.
If the information in your profile is correct your mom at the age of 56 is VERY young and this could be a very long road for both of you.
This forum is great but you need someone you can talk to The Alzheimer's Association has a number you can call any time and talk to someone. They also have lists of Support Groups in your area. 1-800-272-3900 Please call them

Shelia, how old is your mom? Are you her durable PoA? If not you, is anyone? Has she ever been given a cognitive exam by a doctor so that it's in her medical records? This is info that will help us give you best suggestions.