First off, let me say that this behavior can and probably is inherited. My daughter DID NOT grow up around her father at all and yet as she grew up I spotted so many of his habits in her. I KNOW they were inherited because he was never around him to learn them.
Now on to the tornedo daughter. My GD was the exact same way. Her room was a nightmare all the time! I think learning to clean your room is just that - a learning process. I usually go in once a week or so and really clean after she's "cleaned" and she is getting better so I don't have to do nearly as much as I used to.
As for her being destructive to her things during "quiet time," I would change that up and have her keep her door open during this time so you can monitor and make sure she is staying on her bed.
I would also take all of her crayons and other writing materials. For the next 2 months, she would have to ask to use one, would get ONLY one at a time, and has to return it to me as soon as she's done. She now has to earn your trust with a writing instrument.
As for the shoving under the bed, just look every day. I remember one time my mom making me get under my bed and stay there for 10 minutes because I shoved something under it. Let me tell you, I NEVER did that again. Not saying you should do that, just saying it's what broke me.
As for the destruction of the other things, I would simply carry out your threat. Don't buy her anything else. Don't say anything to her, but the very next time she asks for something, very matter-of-factly tell her sorry, you aren't getting anything new until you learn how to take care of what you already have. AFter a couple of times of being denied new things for this same reason, she'll get it.
Good luck!