I fixed the rim, patched the inner tubes, and reassembled the front tire. I’m very happy with my tires: not a single puncture yet… but they sure are a pain to get on and off!
I also investigated the headset issue. I disassembled the headset, which meant a lot of little parts everywhere (note to self: next time, support the fork before loosening the headset, or do it with somebody else helping). It turns out (grr!) that the noise was not coming from the headset at all but from the plastic-on-plastic stretching of the rear brake cable casing on the top brazon of the downtube as I turn the handlebars and stretch the casing. Why that should be happening now, I don’t know. Maybe the cable got stretched during transport. And of course, my little headset dissection, with the handlebar hanging by the cables, did not help.
I had a hard time getting the headset back together by myself, but I figured it out. I think it’s fine.
Next up: fixing the seat post (I broke a screw on Sunday) and realigning the rear tires.