Wage cuts hurt morale. Effort is partly about incentives and partly about trust. The UC Berkeley pay cut. In particular, if the equilibrium real wage falls, adjustment will be more painful if nominal wages fall and less painful if the price level rises. Menu costs probably pretty minor.
Prices do decline in many cases.
Money illusion, in contrast, seems to be a major force. Nominal wages hardly ever decline. People eventually catch on about inflation, but it often takes a while. Keynesian AS Both menu costs and money illusion change the shape of the AS curve in the short-run. Instead of vertical, it becomes upward-sloping.