This is not a requirement. One thing to keep in mind, if there is a nest, they will breed and lay eggs. If you do not want babies, do not buy a nest. You can buy a nest from the pet store. The most common for Zebras is the enclosed wicker nest with a front opening hole.
Newspaper works well for a floor covering.
It is cheap, collects the droppings, and is easily removable. The birds also like to play with it. Other animal bedding materials can be used. Crushed corn cob and pine shaving are a couple. I would not recommend aromatic cedar shaving.