Why do non-binding price ceilings have no effect? My question seems trivial but I have a particular issue I don't understand. If there's a price ceiling above the price equilibrium, the customers can still use the ceiling as leverage when negotiating, and thus negotiate a lower price. Then the ceiling does have an effect.
You are completely right. If in your setup negotiating is an important feature of how prices are set, then even a non-binding price ceiling matters. Search models in labor economics became popular for showing a feature like this. Not with a price ceiling, but with a price floor. In particular when we study minimum wages. In the labor market, the assumption that wages are negotiated is reasonable and then the level of minimum wage affects equilibrium even if it is non-binding. 

The conclusion that a non-binding price ceiling has no effect on equilibrium stems from the assumption that markets are competitive. Prices are not negotiated, they are simply posted and people can either buy or not buy. In that case you should not expect a non-binding price ceiling to have any effect.