I believe that the checkerboard that indicates Vmo/Mmo on the speed tape with slats and flaps retracted drops down to represent the appropriate VFE speed when slats or flaps are extended. For certain the overspeed clacker resets; I can't 100% confirm that the display resets but I believe it does.
The low speed warnings - both the "Low Speed Awareness Indicator" (green line, FAA aircraft only) and the Low Speed Cue (checkerboard) are driven by Stall Protection System parameters and are adjusted based on the current "aerodynamic reality". They can't, however, adjust for things the SPS doesn't know about, so a contaminated wing (which will stall early) isn't catered for, nor are certain fairly obscure failure cases. For some other failure cases, they may be set to conservatively cover all possibilities (if the slat/flap control units all fail, the SPS won't know where the slats and flaps are, so will use a default, conservative, slat/flap position. (Where a failure case does invalidate the indications, the failure procedure should provide guidance)