Can you ship books in a shoebox via the post office? Or do you have to gift wrap it first?

Boxes do not have to be covered in paper.

For a shoebox, I would make sure that you put tape around all three dimensions of the box to hold it together (not just taping the top to the bottom).

Also, you will want to make sure that the books can't bang around inside the box, because as the box gets tossed around by the PO, if they can move around, they might rip open the box from the inside. So, crumpled up paper, or plastic grocery bags, are what I often use to take up the extra room inside. (or, if you have those air bags that places like Amazon use, you can use those too).

And if there are any other shipping labels visible, it is often best to cover them, or at least black them out with marker.

I have received mail in shoeboxes before.  Like Sara said, make sure there is a lot of tape to hold it together.  And yes, definitely package it well because things will move around in the box a lot!

Yes, especially if it's one of the sturdier shoeboxes.  I've mailed book multiples in those before without a problem.  Just tape well, covering any outside holes, and slap the label on top.  It doesn't need to be wrapped.

Yes, you can recycle boxes to ship USPS.  They do not have to be covered in paper unless the box's original content was something banned from being shipped via USPS, such as alcohol, lighter fluid, etc.  If it cannot be legally sent USPS, then recycled packaging must be covered.

When recycling, deface any existing USPS barcodes with a perm. black marker...other barcodes do not matter, just USPS.