Hi, Baby-Daisy,
I wouldn’t bother with the cat cube. It’s not a bed for a cat or a bunny, and doesn’t have any real benefit (but possibly does have problems) for a rabbit.
The basket, while it may not serve as a bed for your bunny, would certainly be welcome as a chew toy, container of hay or possibly work if lined with fleece bedding.

Please consolidate your other bedding question into this post. It is really the same topic posted elsewhere. I will copy your question from there to here and delete that post, OK?

Baby Daisy also writes:
12/15/2012 11:46 AM Delete Edit Quote Reply Alert
How can I make a rabbit bed that is safe for my rabbit if she chews on it, and how can I make it so it does not tare up eaisly?

get a small cheap bed and put it inside a thick pillowcase..

If your bun is a chewer, any type of fabric bed can pose hazardous. My buns have blankets covered by upside-down cardboard boxes with a door cut in them. This is where they love to sleep. They don’t chew the blankets. I had another bunny who did, so he was never able to have blankets. I gave him shredded paper in a box. I think buns prefer “cave like” beds since this is what feels natural to them. It gives them a sense of security. Can’t go wrong with a cardboard box! I had tried cat beds before and they just got ignored and peed on.

This is Olivia’s bed. 

The door used to be just big enough for her to fit in but she loves making it her own. She’s a real interior designer. She’s not a blanket chewer, but for anyone giving blankets to a bun for the first time, keep a close eye on them. Ingesting fabric is very bad.