Say, for instance, we need to separate the BitsyX from a Sharp LCD like the one provided in the development kit. Does anyone know how far one can make the cable between the two before running into problems?
I wouldn't go any more than 18 inches/50 cm, and that only for prototypes. The pixel clock is up in the 15-50 MHz range for many of these displays. Where you'll especially start getting into trouble is with EMI, but longer cables can also start causing distortion or interference on the display.
LVDS is your best bet for running display signals any distance. The EMI issue goes away, there are fewer wires and you can run the signals several meters.