Going from 3 way switch to a regular switch
The right 3-way (center switch) has neither hot nor neutral.  The three wires present there are: 


messenger 1
messenger 2
switched-hot.  


Since you have neither hot nor neutral, that far 3-way is dead as far as feeding power off it.  You have to go back to the left 3-way.  

You also have a problem there paralleling two 12-2's, despite what the illustration shows, you are not allowed to do that.  

I would start the whole thing over with smart switches which use radio waves for 3-waying, and simply straight-line hot and neutral through all boxes.  


Switch-1: Smart-switch master that supports radio comm with a powered (or battery) remote.  
From switch 1 to lamp 1, black=always-hot, white=always-neutral, red=switched-hot. 
Inside lamp 1: TOP 12-2 is passthru: black goes to always-hot, white goes to always-neutral.  BOTTOM 12-2: Black goes to switched-hot (red wire) and to lamp.  White goes to neutral and to lamp.
From lamp 1 to 2:  TOP 12-2: passthru hot and neutral.  BOTTOM 12-2: switched-hot and neutral for lamp 2 only.**
Inside lamp 2: TOP 12-2 passes through.  BOTTOM 12-2: black is switched-hot to lamp.  White is always-neutral to lamp only, do not parallel with top 12-2.  
From lamp 2 to switch 2: black=always-hot, white=always-neutral, red=capped at both ends. 
Switch 2: Appropriate smart-switch remote. 
From switch 2 to switch 3: Black=always-hot, white=always-neutral. 
Switch 3 and past: normal switch circuit. 


** This arrangement is very important because it assures currents are equal in all cables.  You can't randomly parallel two 12-2's if you need 4 wires, the diagram you linked is wrong.