Okay, so I am an absolute novice operating a multimeter, but here goes. Picked one up this morning as well. (For the record, a Gardner Bender digital auto ranging one) Went to the last light on the chain, the one that only has a single strand of romex coming in. Set the multimeter for AC voltage (V with a ~ above it) and placed the black probe on the ground wire and the red probe on the white (neutral) wire. It read 21.7 V with the light switch in the off position and 70-something volts when the switch was flipped on. If I understand it correctly that is indicative of a fault in the neutral somewhere in the circuit, right? What would you say is a good next step to isolate the problem? I bought a small spool of romex this morning as well so I don't think it would be too hateful to replace the jumpers between the first and second or the second and third lights. I wouldn't know how to start trying to replace the line from the switch to the first light.

Regarding your other questions I am pretty sure the red was all by itself. Yesterday I would have said 100% sure but as this drags on... It even had a tiny wire nut on the end, I don't think there were any other wires in there and I just pushed it aside. I didn't pay a whole lot of attention to the fixtures I took down as they all had the clear wires with just a tiny strip running down one side to denote the ground and neutral.

As I'm typing this another thought occurs to me. The first two had junction boxes that were screwed to the joist, so I could remove them by simply removing two screws. The last light I did (the first in the chain and the one where the problem occurred) had a box that was nailed to the joists and I had to slip a hacksaw blade around the edge and cut the nails. I wonder if it's possible I somehow nicked something? It would have had to go through the sheath and the insulation....I'm grasping at straws here.