Hi Twopoodles
You have my sympathy, caring from a distance has it's own problems, and for you in Australia it must be even more difficult. My heart goes out to you and your siblings at this difficult time.
It's very frustrating when you put care in place and the PWD is resistant as they feel they are fully capable, when in reality they do need this support.
My sister and i tried to do our best for mum to keep her at home for 2 and a half years after dad died, but it was really hard going, as mum allowed the carers to do very little.
in the end we had to move her into care as it was becoming dangerous for her to live by herself at home, and she was getting increasingly lonely.
I hope that you will be able to sort things for your mum whilst you are staying over here, and maybe she will accept moving into care before you return to Australia so that things don't have to come to a crisis once you're gone and there is no-one close by to keep an eye to her.
If you haven't done so already it would be good to get POA in place whilst your mum still has capacity to agree. You, and your 2 reliable siblings could all be on together, as "jointly and severally" , so you don't all have to be together to action things. We found it so useful for paying bills etc.
Mum didn't have pets so i can't advise about the cat, but think you would be best to re-home it. I am sure your mum would miss it, but i have heard others who have got a "toy" cat for their PWD, they can be very life like and others have said they bring great comfort.
Wishing you and your family all the best with looking after your mum.