Both the single lilypad pcb-mounted LEDs and the lilypad RGB pcb-mounted leds have limiting resistors already fitted, so that’s one less thing to worry about.
There’s no need to connect the parallel white LEDs to the lilypad twice per electrode, once you’ve made the connection, you’ve made the connection. (Are you sure it’s supposed to connect across two output port pins, rather than one output and ground, or one output and +ve (whichever way round you end up using the LEDs?)
The diagrams for both ways of connecting the RGB leds are the same — they’re both in parallel.
The lilypad light sensors are also fitted with the correct resistors for a 5Vdc power supply.
The AAA psu puts out 5Vdc (it’s quite a clever psu). However, you’ve got the connections a bit wrong on it. The AAA psu actually has three negative terminals and only one positive terminal!