Do I need a 3 wire cable between light fixtures?
You only need three wire between the fixtures if the switch is on a branch off from the lamps 

supply =2+e= lamp =3+e= lamp =2+e= switch


but new requirements for smart switches require 3 wires to the switch in that setup now

supply =2+e= lamp =3+e= lamp =3+e= switch


That setup is frowned upon these days because CFLs can blink when the switch is off.

You have the supply going direct to the switch (which is the preferred setup) so you only need 2+earth going from the switch
to the lamps

If any of those boxes are metal the earth wire should connect to the box. if not just join it with a wire nut as you show.