If the Chief Justice is in the majority then he writes the majority opinion.
Otherwise the most senior member of the Court, who's in the majority, either writes the opinion or assigns it.
Perhaps an actual legal expert can answer your second question but my understanding is that based on the below quote is that if you already voted not to hear the case but the case is being heard you really should try to at least examine the merits and not just continue to say that the case shouldn't have been heard.
The suggestion that the writ be dismissed as improvidently granted raises a recurring problem in the administration of the business of the Court. A Justice who has voted to deny the writ of certiorari is in no position after argument to vote to dismiss the writ as improvidently granted. Only those who have voted to grant the writ have that privilege. The reason strikes deep. If after the writ is granted or after argument, those who voted to deny certiorari vote to dismiss the writ as improvidently granted, the integrity of our certiorari jurisdiction is impaired. By long practice-announced to the Congress and well-known to this Bar-it takes four votes out of a Court of nine to grant a petition for certiorari. If four can grant and the opposing five dismiss, then the four cannot get a decision of the case on the merits. The integrity of the four-vote rule on certiorari would then be impaired.
