For example, if you deposit the bee as a collateral, does it still produce $honey or not? If so, can the bee pay itself off with it?

If that were possible, you could use the money you borrowed to buy the NFTs from the projects listed on Honey.Finance, so the other projects would be covered by us. And all the money (USDC or Sol or Honey) you borrow stays in the Hive with the Bee's