If no one bids on a rockstar, and the player who started the auction passes, who starts the next auction? Does the auction choice pass to the player's left or stay with the same player?

Rock on.

The person who started the auction must either pay an envelope to put the rockstar on the bottom of its deck or pay an envelope to take it (assuming nobody else bid on it). You cannot put a rockstar back on the bottom of its deck 4 times. The next person to start the auction is still the player to the left, no matter what happened.