How do I properly go about asking mom, and successfully having her, get rid of her cat?

Just take the cat and get it fixed. As mentioned before, there are low cost spay services available. Usually you take them back home the same day. Just tell mom the cat is hiding.

In your shoes, when mom was asleep, the cat would disappear to an animal shelter and never be seen in the house again. And if mom asked, I’d react just as I did when I had toddlers and they missed something like a pacifier that I’d gotten rid of, just a puzzled look and a quick “wow, wonder what happened?” No cat needs to be where they aren’t wanted and no human needs to live in misery

I know if you have discussed this with your mother? If you have, can you please tell me her response to you?

I would go to your nearest shelter. Many have free spay/neuter programs. Often these are meant for feral cats that people trap, take to be spayed, and return to site. If you want to avoid the cost of checkup, vaccine, and etc. then I would say this is the way to go; at least it is worth a try.
I would simply then take the kitty and get this done. Tell Mom you did it for the cat.

Now we come to the bad part. This kitty may not stop marking just because she's spayed. For a while she will still have lots of raging hormones. She may or may not stop eventually.

I agree with others that it makes a difference here whether you have entered your mom's home because you require a place to stay for a while or you entered her home because she requires your help. The latter would give you some rights to sleep without being urinated on. But the former reason would mean you realistically don't have a lot to say about how your mom lives, just as you would not want her judgement on how you live.

Good luck, and welcome to the forum. For the kitty's own good I hope she gets spayed.

Convince Mom that both the cat and dog need their teeth cleaned.
Take them to a vet who knows beforehand that the cat needs spaying.
Ask him to give his advice on spaying the cat, have him be persuasive
and counsel mother.
Have the vet keep kitty overnight (for some reason), and let Mom decide.

Help her pay for it?

Do both the cat and dog receive the needed vet care?

Check with the Humane Society in your area and explain the situation about your mom's cat. Tell them that you need to have this feline spayed like yesterday. The spraying could be health related or behavioral. Tell them that you mother is elderly and on a fixed income, and ask if they have any programs that will spay the cat for free or for a small fee.

As for the spraying, settle this pee monster in a small room with a litter box, food, and no surfaces that are soft other than her cat bed. Talk to the vet on what measures you can take to re-acclimate the cat to the litter box.

I have a pee-er too with my female. She is spayed and would use my rugs. Finally, I threw out the rugs. She has a bucket that she pees in. (No joke) It is a small dish pan thing that I use to mop my floors and I left it in the bathroom, but she has taken it for a pee pot. I have yet to see how she uses it. She doesn't like to use the boxes after the boys. I don't blame her though. Like most males, they can be quite gross when it comes to bathroom habits.

Your girl may not like the brand of litter. First though, get this little girl to the vet asap!

I have owned 4 cats in a 30 yr period. 3 males and one female. My female was the sprayer and fixing her did not change it. I could not get rid of her, she lived to be 16. Once they spray, there is really nothing that will cure it. She wrecked a entertainment unit I loved. My furniture. My rugs. My porch. Not to mention other things. When she passed I replaced everything. In our 70s we will not have another cat. Nor dog, neither of us really wants anyway.

What I would do is take it to a no kill shelter. They will spay/neuter it and put it up for adoption. Tell Mom the cat got out of the house. Hopefully spaying it will correct the spraying. I am so with you because the smell of urine from a unfixed cat is awful and very hard to get out of stuff. Any rooms we could shut the door were shut while the female was alive.

I won't say to put yourself in your mom's shoes. I would say:

Find your own shoes and put them on.

While you do not say anything about your mother's mental health, age, etc., she may have cognitive functioning challenges/issues due to dementia. There is no logical or health 'reason' for your mother to believe / insist as she does. I suggest your mother may need to be medically assessed. It will benefit you to understand her cognitive functioning/abilities.

If I were you, I would get the cat spade and not tell her - if telling her will cause problems between you - and from what you say, it will.

You need to take charge; own your power; and do what is in both your - and the cat's - best interest. And, your mother's too although she won't acknowledge that.

The other possibility might be for you to leave.
Do you have to stay in the home with her and the poor cat in heat, howling?

She'll be mad at you and that is okay.
Do what you know needs to be done.
Deal with the fallout later.

When you take your power back, her 'expressing anger and worse (?) will not affect you as much as it might have before you 'stood up for yourself - and the cat. It is cruel to do this to the cat. You have to manage the situation. Your mother clearly has no compassion or consideration for the cat. It is up to you to do what is (also) in the best interest of the cat.

Gena / Touch Matters

Look for a free spay/neuter clinic and get the cat spayed.

Shelters are overrun at present, and if surrendered to a shelter, the cat is likely to be euthanized unless it’s a no-kill shelter.

1. Take the cat to a vet and get it fixed.
2. Don't live in Mom's house.
3. Learn to live with the cat and any other unpleasantness of living with Mom.

You don't give us much to go on, other than complaining about Mom's cat,
Which doesn't seem relevant to this forum.

Is there something more? Anything more relevant to those of us caring for a loved one with dementia or a loved one with serious physical limitations?

We all strive to help one another out here. I know my answers sound flip, but, you're complaining about a cat. At your mom's house.
I don't have any experience or expertise in dealing with that.

If my simplistic answers don't help, perhaps you can share more about the situation. Or, refer back to #1.

Years ago I had a female cat that I didn't want to have spayed because I was concerned with her being overweight after spaying. (This was dumb on my part, of course.) The cat stayed indoors only, so no risk of kittens. When the cat was in heat, I tranquilized her. I was fortunate because I had access to and permission to use major tranquilizers that I could give her by injection--only needed to do it a couple of times a day. Since I was giving animals injections in my work, it was not a problem for me or the cat--took about 15 seconds. There was also never a problem with odors or spraying. If you absolutely cannot get the cat spayed, I would ask the vet for a major tranquilizer to give orally. If someone shows you how to do it, it is very easy to give cats pills! I don't know what would happen with the spraying problem because I've never seen that in a female, only in a male, neutered or not! Female cats may emit an odor when in heat, but I don't understand physiologically how they could spray!

I later learned the folly of my ways, and I had future cats spayed as young as recommended and advised others to do the same. There is a good health reason for this, too: it lowers the incidence of breast cancer. Perhaps you could ask the vet and then tell your mother that the vet said the cat needed spaying to prevent breast cancer. If she still loves the cat and can still reason at all, maybe that point would convince her. If not, I would do what others recommend and have the cat spayed anyway.