LED Lighting not working
LEDs don't act like ordinary incandescent light bulbs.  When lit, they have a fairly constant voltage drop that varies slightly with current.  White LEDs have a typical forward voltage of about 3 volts.
The single #5 LED will draw enough current from the 9 volt battery to pull the voltage down to about 3 volts.  The other LEDs which are connected as two LEDs in series, will require about 6 volts to light, but since #5 is holding the voltage down to ~3 V, they will not light.
When you remove LED #5, the voltage will rise to allow the other LEDs to light - but the 9 volt battery cannot deliver enough current to light them to full brightness.
LEDs should never be connected directly to a voltage source - they always require something to limit current (normally a resistor).
You should connect one of the resistors you show in series with each series pair of LEDs.  That resistor is probably 100 Ohms - definitely not 100 MegOhms!