Does the concept of Nash-equilibrium conflict with the concept of market equilibrium in the lemon market
So this seems to be a known issue. Quoting from the Wilson article of 1980, 
The Nature of Equilibrium in Markets with Adverse Selection:


  Using a variant of Akerlof's model of the used car market, we examine the equlibrium of the model under three distinct conventions: (1) an auctioneer sets the price; (2) buyers set the price; (3) sellers set the price. Only in the case of the auctioneer is the equilibrium necessarily characterized by a single price which equates supply and demand.


Thus the answer to the question is a partial yes. Given some (most) variants of the lemon market the two equilibrium concepts do not result in identical outcomes.