I sent an email and received an auto-reply stating that the recipient is out of the office and will be on duty after a week. Does it mean that my email has been read by the recipient?
No. Out-of-office autoreplies are sent by the receiving email server independently of any user action. That's what makes them automatic replies.
It does mean that the mail has been received by the recipient's mail server, but does not directly indicate anything about whether or not a human has seen or read the message. (It does, however, imply that they probably won't see it until they return to work.)