3-way switches and independent lights
OK first, nobody has a chance to figure out that Egyptian Spaghetti (black red white bronze = Egyptian flag).  So let's hit up the electrical supply house for some tape, and re-mark the colors so we can see what's going on.  

Normally I use the same color for both messengers, but in your case, it matters. 



Normally, a 3-way works by a 3-way connecting always-hot to one of two messengers (the other is dead).  Then the other 3-way connects one of two messengers to switched-hot.  If the switches agree, switched-hot is energized, and the lamp gets switched-hot and neutral, and lights when switched. 

Now, you just horked on another lamp and 3-way, as if this was Insteon or something.  Look what you did.  

The original 3-way circuit still works normally.  Bulb B gets always-hot and a messenger.  No neutral in sight - that's a big problem.  If switch 1 agrees with 2, you have 120V on both sides of bulb B, and no light.  

If switch 1 is the minority report, then it's talking to a dead wire, and no light. 

But what happens if switch 2 is the minority report?  Follow the loop:  

Always Hot - Bulb B - switch 1 - dead messenger - switch 3 - bulb A - neutral 

You have put bulbs B and A in series.   Now if bulb B is an efficient LED/CFL, and bulb A is an incandescent, A will barely light.  



How do you fix this?  Easy.  Get switched-hot and neutral to  bulb B.  Notice that neither one is anywhere near bulb B.  The wires in box 2 are all the wrong wires.  Literally every wire except the two you need.    So you need to run a cable from bulb A, because that's where switched-hot and neutral coexist. 

If you want all 3 to work as multiple control points, then you need to fit a "4-way" switch for switch 2.  A four-way takes two messengers on the left, and two on the right, and either passes them straight through, or swaps them.  You would feed always-hot at switch 1 instead of 2.  

Or junk all this, and go Insteon or other smart switches.