First now take a deep breath.

Even without a realtor hot on your tail, this time would be stressful. Please take ONE day at a time.

There are benefits to your current landlord selling. #1 it seems that they are really irresponsible and inconsiderate, and not in control of their own finances. #2 selling the house would stop the showings. #3 maybe the new landlord will be able to help you with things that the old one couldnt/wouldnt.

Make a safe room for the cats to be tucked into for when the showing is coming. Even if that means they get tucked into a carrier for the hour that its happening. Throw a towel or blanket over the cage so they are in a cave. They like that. If you pop them into a room that has their water/food/litter in it, this will prevent them from running under foot with strangers. its very stressful on animals to have strangers in and out of the home. Put a sign on that door to Keep closed or keep cats IN that room please.

Its best to try to make friends with this awful realtor. I know your instinct is to scratch her eyes out, but if you can get her more sympathetic, perhaps she will be more reasonable about showing up. If Voice mail is an acceptable way to contact you for the 12 hours, then...unhook the VM from the phone! This will force them to have to speak to you personally. If there is no way to leave a message, they must find another way. Be VERY nice to her, and try to engage her on her level. i know you want to bump her with your car, but it will only make it worse to make her your enemy. Ask her things like "i know you are working hard to sell this place (a compliment to her) are there things I can do to make it easier? Were there comments on things that can be corrected? (this makes her think you want to help her, even if you have no intentions of doing any of it.)

Take care of YOUR family. If that means unmade beds, or dirty litter...then leave a note explaining your situation on the counter, and beg for understanding. I bet most folks she is showing this house to are not as intolerant as she is.

Come here and vent it really does help.