If you follow the datasheet, you’ll understand what you might or might not want to do with RW. Remember, I’m telling you about what the Gadgeteer code and module do. If you want RW, connect it, and figure out where to amend the code. Otherwise, the code as-is assumes RW is connected to GND as per schematic.
Backlight is on pin 8 on the Gadgeteer socket. In the code you pointed to, is:
backlight = GTI.DigitalOutputFactory.Create(socket, GT.Socket.Pin.Eight, true, null);
What more do you need there?