Progress.
I noticed that it didn't have a kickstand when I unloaded it at my house, so had to strap it to the underside of my deck
Picked up an XL125 one off ebay for $15.
Petcock was leaking so I bought a rebuild kit. Turns out the 4-hole washer was damaged, so hopefully a new one will fix it.
Also had some cracks in the main tube, but they're above the reserve tube so I'll leave it be for now.
P.O. said he'd put some Metal Rescue in the tank to clean it out, but there was still a decent amount of rust in there, judging from the state of the petcock and the last bit of gas that came out. I used a big fender washer, a small rubber o-ring, a regular washer and a piece of intertube and made a lil' plug for it. Leak-free! Treating with EvapoRust now.
Carb needle was set in the wrong position - should be in the middle.
Carb and bowl cleaned. Carb was also missing the pilot screw spring. Float level was WAY off. All back together now with new rubber.
Wiring was janky so I re-did it last night. New connectors, and (hopefully) working kill switch.
Replaced the clutch perch with one from a KZ400 - adjuster threads were stripped. Still needs some new tires & a tuneup. This weekend I'm going to try and make a heat shield out of a cutting board, kinda like this.