Who goes next after a double auction card with no auction?
From checking the rules in front of me the next player clockwise would be next auctioneer.

In the 'running and auction rules' it states :-


  When an auction ends, the player who made the highest bid pays the
  auctioneer the amount of the bid.  If the auctioneer made the highest
  bid the money is paid to the bank instead. If you win an auction, take
  the painting and place it face up in front of you. (...) The player to
  the left of the last auctioned now offers a painting card from his
  hand for sale.


In the double auction rules it states :-


  If it goes back around to you, with no player offering a second
  painting, then you get the double painting for free.


Whilst the word 'won' isn't in the rules that player did win the auction and paid nothing to the back and place it in front of them so the next player clockwise becomes auctioneer.  

Also I think the idea that the player who played a double auction card had had nothing goes again is unlikely.  This kind of exception to a rule is likely to be explicitly clarified so it feels even more likely that play would just go clock wise as in all other parts the auction rounds.