When you authenticate your computer will you ever have to re-authenticate the same computer in the future? (In other words, does the authentication expire?)
No, you won't. The only reason you should ever have to re-authenticate is if, for some reason, you think that the QR code or 16 character code got out of your control and might be used by someone else, at which point they give you a new code (they should, theoretically - I haven't tested this out yet, but not changing the code each time doesn't make any sense).
You won't have to re-link the authenticator on your computer/smartphone again, no, but you do have to enter the short code generated from the authenticator once a month.
So it's the same code every time? Or will that have to be generated every month?