I'll put this here and leave, because I know it's not going to be popular on Mumsnet - most people here seem to have decided that autism is the worst thing ever and you can't trust a person with autism and Aspergers to look after themselves, anyone else or even treat other people with decent respect. If you read some of the threads on here, you'd come away with the impression that we're unfeeling monsters and energy vampires.
In fact, as it's a post complaining about the autism bias, I expect it to be deleted fairly shortly.
But while it's up - OP - YABU, mostly because it's got nothing to do with you. You're not her carer, you talk about her whims and obsessions with distaste and you've overridden her based on your opinions of her competence and you've made derogatory comments about your sister to another person.
She might not do well with the dog, Fine, in that case, report it as soon as you're genuinely concerned about it's wellbeing.
But until then - why are you judging her guilty of a crime and limiting her behaviour when she's hasn't done anything wrong? She's ASD and you believe your mum is gullible and naive and your sister is unable to look after herself - but maybe, just maybe, they might know something you don't? You don't live with her, you just seem to see the worst in her and you think any other opinion isn't realistic - sounds like you've already decided she's not going to be capable of much.
FWIW, I'm ASD, have similar obsessions - including animals - struggle with anxiety and depression - I also still have toys - but having an animal was a good responsibility for me and I grew into the role and it gave me some order, discipline and a routine that I desperately needed. I now realise that my first dog shouldn't have been a GSD but, despite the hard work and shock to my system, she was well looked after, I adored her and she lived a long and happy life. Since then, always had animals - even fostering rescue dogs - I'm grateful that the rescue I worked with had more faith in my abilities than you do in your sister.
I also had children too - despite being told that was a bad idea - and surprisingly they didn't die or savage anyone either.
What are you going to do if she gets pregnant, OP? Should you be calling up her obstetrician and telling him that she needs an abortion because she can't deal with a baby?
You might think she's child like and unable to cope but sometimes people step up to the plate and learn by the experience. She may be one of these but it sounds like she's never ever had the chance to learn.
I'd suggest that, if it bothers you so much, you just don't get involved. You don't own her, you don't know that she'll be incapable, you've just decided you can't trust her to do anything. How much are you going to stop her doing in her life? Why not call up any potential partner she's seeing and tell them that she's not fit for a relationship because she's so gullible and easily fooled.
Poor girl, her confidence will be rock bottom and now you've decided to humiliate her in front of a stranger and make a point of how incapable you see her.
I can understand why she's depressed.