If you program your own keys into the Yubi, then you know them and can archive them for reprogramming on another device. You can do this with the Yubi Personalisation tool [1] for a few modes the device supports.
Eh. Hence why I said it like I did. In most cases, the device generates the secrets. And that's how it should be done, it guarantees that they can't be compromised easily (vs if someone compromised wherever you backed up those keys to).
Sure. There are also other instances where the Yubis keys maybe exposed, such as when using their OTP protocol which requires the keys stored in a validation server (either theirs by default, or your own [1])
Pretty much every online service that supports it has a backup 2FA method that you are required to set up. I use Google Authenticator on my phone because my phone doesn't have NFC.
EDIT: I mean you can't copy credentials off in most cases, like this one. Credentials can be replaced.