I would phrase it less as asking her to lock her dog up, and more "hey, since he's not feeling well right now maybe we could set him up in your bathroom with a blanket and a bowl of water? Keeping him in a smaller area while he's sick will help him relax a little easier (dogs are den animals by nature) and keep his water in closer proximity (which will prompt him to drink more and stay hydrated since he doesn't have to look for it)."

And equal for all is what you want! You don't want her to feel singled out and you guys don't want to be (and shouldn't be) putting yourselves out more than anyone else.

Because y'all have let things go for so long, as I mentioned you've got to keep it light and like you're suggesting things for her best interest (reality is it benefits you as well). Once you've planted the seed you can start getting firmer with, "hey these are your dishes in the sink." or "your dog seems pretty stressed out running around while you're not here, have you considered crate training?" etc. But you have to lay the framework, if that makes sense?

I would consider your relationship with her separate than the other roommate's. If she feels like everyone is against her she's going to be pissy and miserable to deal with (and frankly, wouldn't you be?). She seems like a hot mess, to be honest. Realistically though you just need to make it to the end of your lease. Since going out with her doesn't seem feasible, why don't you guys start a netflix show together? Like have a girls' night and every few nights you watch an episode (or spend a day bingewatching now and then). Like I said, getting to know her more (and her you) will help her relax and be more receptive to criticism.