I have to say I agree with OutNumBird.
Can't really comment on whether your siblings are being selfish - presumably they live and work abroad, and don't just spend all day sunning themselves on beaches?
I'm always a bit wary when people talk about family who have moved abroad as being 'selfish' because I've seen a few situations where family members try to guilt trip others who move away. Anyway, only you know your specific situation.
The other thing is, in reality, if you and your two siblings did genuinely try to divvy everything up fairly, then tbh, it would mean 3 people probably not being able to hold down a full time job because they'd have to be available to be called on in times of need. The harsh reality is that most people have to be getting on with their own lives and this might mean moving where the jobs are etc.
One of my siblings does a lot more for my father than I do, but the reality is that this is the only sibling out of 3 of us who lives near my father - and the reason for that is dead simple - he has a very higly paid job and can afford to live in that area, whereas my other sibling and I had to move away to be able to afford to live! So while I sometimes feel a twinge of guilt that my brother does more, there's no point dwelling on it.
I also think your mother is being really selfish in being so uncompromising about the pets etc. If she's dependent on others for care, then she ought to shift a little bit on this.
I agree that it's time to call in Social Services. I know that's hard when it goes against the grain, and some families have a big thing about wanting to keep care within the family. But I believe there comes a point where it's not all about the person being cared for - other people have needs too.