I'm noise sensitive so I sleep with a fan. I have two questions:
1) Who's problem is it? It sounds very much like it is her problem not yours. She knows she is noise sensitive but rents a downstairs apartment. She knows she has trouble sleeping but doesn't see a sleep doctor or play a sound machine or offer to. It you a nice thick rug in the color of your choosing to go in the room above her bedroom.
2) did you sign a lease? Did you have permission to have a pet in the lease? If so again this is not your problem. The lease says you can have s cat, you have a cat. The end. That said as soon as you can , just move. Do you really want to live with that crazy dysfunction?
*i would write a certified letter or fax to your landlord explaining the situation and complaining about being bugged with all her complaints! I would also formally request all correspondence with the super/landlord/other tentant be in writing. Honestly I rent a room with thin walls. I'm a night owl but I make an effort to be quiet if it is past 11pm and try to be quiet from 9 pm on. You are way nicer than I am. I would get a rug (and let her buy the padding if she wanted) but I would leave toys out for my cat.
Just remember you can't use logic on crazy and it is a waste of time to try. Stick to your legal rights (feel free to research them in your area.) The sooner you can get away from the crazy the better. (If you just leave you break the lease and will have to pay a fine or keep paying rent until the end of your lease.)