OMG I have a brother, and he lives in Australia?!?!?
I'm sorry to make light of your situation, but I have a mother that rivals that. I learned a long time ago to not take it to heart, she is just bitter, mean, and hateful. She doesn't even realize it most of the time.
Just have faith that Karma will greet her one day too. If you can't limit your interaction with her then just try to not engage her.
I can honestly say the older I get the easier this becomes. My loving mother would be shocked if she knew I felt this way, but this woman is just F'ing mean.
You want examples, here's a few charmers from the past week.
My father willing donated firewood to us last winter, because we didn't have a chance to get down in the woods and get our own. When I approached him about having a day for the 4 of us (my H, me, mom and dad), he tells me we can just have more. 15 minutes later, I get a 15 minute tirade, how it's not right for us to just use "their" wood, and that we are selfish, and that if want wood this year, we are going to have to get it on our own, because we "refused" to help them last year.
True story, we were called at 9am on a saturday that had a Marching band competition, a Cheer thing, and dinner with inlaws, to help that very day, when we said we weren't available I guess that means we refused. It's insane.
I don't let it bother me though. I know she is selfish, bitter, and angry. I just pray when she gets dementia that she does a 180 on personality, because sister, she will be impossible to deal with like this.