Baffled by 3-way switch wiring
Electricity travels in loops

In utility electrical, every hot must have a return wire partnered with it.  They must be in the same cable.  All the currents in each cable must be equal: the current coming down as hot must return up as neutral. With 3 or more wires, the currents added together must equal/cancel out.) 

When a neutral is disconnected, it will light up as hot. Electrons are trying to return, and they can't. So this "floats" the neutral up to 120V.  This is a really good reason not to bootleg ground -  nor use 3-prong dryer/range outlets, which "legally" bootleg ground. 

Get some colored tape

3-way circuits will make you ape crazy, because the physical topology of your wiring, and the equal-currents rule, forces some strange routings sometimes.  Here's what is happening electrically, and note each wire has a different purpose, and in this diagram, each purpose has its own color.  



There is no reason to distinguish travelers from each other; they are interchangeable.  

Of course this does not match either your wiring topology or your colors.  I would make it so using colored tape or shrink wrap.  

So assuming your /3 cable rings out alright...

Figure out how you want to wire your 3-way circuit.  I would take always-hot down to the remote 3-way and bring two messengers back.  Black is a good always-hot color, so I would tape the red and white wires yellow.  

Your neutral from supply is already tied to neutral up to the light. Done. 

Your hot from supply is easily tied to the /3 cable black.  

On both 3-way switches, the yellow wires go to the traveler (brass) screws.  

Each box has only 1 wire left, that goes to the common (black) terminal.