Has any sin of the prophets (Hud, Salih, Yahya) been mentioned in the Book or Sunnah? And does the absence of mention of a sin for Christ (peace be upon him) mean that he did not sin?

There is no text in revelation that mentions sins for the prophets who are named. The absence of mention of Jesus's (peace be upon him) sin in the Hadith of Intercession may mean that he never sinned at all. Prophets are infallible from major sins and in delivering the message. As for minor sins, they may commit them, and Allah warns them about them, whereupon they immediately repent. The prophets for whom sins were mentioned hastened to repent and seek forgiveness, and they were more perfect after repentance.