While I hesitate to say "All", its safe to say "Most if not all" Altera programmnig cables with will with Nios II, ByteBlaster, ByteBlaster MV, USB Blaster, etc.
About the LCD: Yes, it does work with Nios II on the cylone board. One known issue (which affected things previously as well) is that the pins the LCD connects to in our example designs are shared with the compact flash connector, so to use the LCD you should unplug your compact flash card if there is one present.
The board diagnostic is a good way to check out the LCD operation -- I am surprised it doesn't work. One other thing is to ensure that its connected properly -- it should be plugged in to J12 on the cyclone board with the ribbon cable coming out of the connector pointing "down" towards the bottom of the board, where the user push-buttons are.
Hope this helps!