Melted ground wires together
Reading your question I think you have or had a high resistance joint in your panel on the neutral. In general electrical circuits in your residence wire all loads in parallel. This distributes amperage through each light fixture and the voltage remains the same. 

If you had a loose connection of the neutrals at the panel this actually becomes a load in series with the rest of the circuit. The high resistance at that point would cause a voltage drop building up heat and melting your insulation. It would also show voltage on the neutral before it dropped at the joint.

I can't say that this is the only problem on your circuit but you must clean this up before trouble shooting any other problem that may be on the circuit. Make sure the circuits on the neutral are off then disconnect the neutral from the bus. Clean off any melted insulation, and clean the conductor so it looks "like a new penny". Make sure that when you reattach the neutral to the bus that it is also clean of and burning or corrosion. Make sure the neutral is mechanically and electrically secure to the bus. Then turn your circuit back on and see if there are any other problems.

Hope this helps and good luck.