My mother dealt with depression for a very long time and still does from time to time. I can completely empathize with you when you talk about how your mother says that you and your sister do things "because you don't love her." My mother used to do that all the time.
Honestly, she just wanted to hear us say that we loved her. Yeah, it got old. But she just needed the extra support and the extra loving words. Did she go about it the right way? No. Did it make us feel good? No. But every action has a reason, either conscious or subconscious. This is one of those subconscious ones.
It does also sound like your mom has some control issues, and for that, you might have to use some reverse psychology. Say, your mom says your sister can't move out. You sister could say, "I was really hoping you'd help us decorate, Mom. I really love you sense of style, and this will be my first time decorating a home."
Suddently, Mom is in control again - or so she thinks. You sister has given Mom a task - but has also used the words 'will be,' insinuating that she's moving out whether Mom helps decorate or not.
Obviously you can adopt the situation to meet your needs. The bottom line is, your sister is an adult, and she can do what she wants. Your mom might pout a bit, but... there isn't anything she can do. As long as your sister is paying for everything, she doesn't need you mom's permission. Eventually, your mom will see that her kids are growing up.
With people who like to be in control, you just have to remember this: make your idea their idea. If they think they have the upper hand, you can't lose.
Good luck, hope everything works out.