The language in the job posting says, "Applicants must arrange for these and three letters of reference to be sent directly to." I have already requested three referees to directly email their letters to the professor. Is the protocol that I'm not supposed to see the letters of recommendation? Can my referees request confirmation of receipt of their letters?
The language is clear that the letters are sent "directly", rather than through you. So, the writers send the letters themselves.
Even when the protocol is that you shouldn't see the letters, some writers will show them to you anyway, thinking it is harmless. The intent, is that the writer should be able to say whatever they think appropriate, even if it doesn't flatter the applicant. But you can assume that the protocol is that you don't see the letters in advance. Moreover, you give your assent to that "private" communication about you.
Yes, they can ask for confirmation, though whether it is sent or not might vary.