OK, I figured out my problem with serial connections. I was powering the frankenpill via a laptop USB port. But that resulted in the F3 going into USB DFU mode (which enumerated correctly, but I never noticed), and the serial upload getting disabled. When I powered it from a wall charger, the serial showed up just fine.
The re-soldering was actually pretty easy.
0. Make a note of where the alignment dot on the IC is.
1. Clamp pill in vise.
2. Thread an old guitar string (=strong thin steel wire) near one side of the IC under the legs.
3. Hold guitar string with one hand and start heating the legs starting on the side where the hand is. Gently use the guitar string to pry the legs off. Repeat until side is done.
4. Repeat 2-3 for the other two sides.
5. Clean up a bit with desoldering braid. But not too much: you want to keep solder on the pads, just smooth it out.
6. Put new chip where it belongs, remembering where the alignment dot was. Align carefully with pads. This is actually rather hard. I used tweezers.
7. Gently touch a hot fine solder tip to a corner leg for 3-4 seconds. And then another, ideally (I can’t remember).
8. Repeat for other legs.
9. Check for shorts. I had one, which I fixed by cutting with an Xacto knife.
Besides the one short, I had one other problem: I wasn’t careful enough at step 3, and two legs pulled up parts of the traces. I ended putting drops of solder paste on the remaining bits of the traces hope things would join up and bridge, and it seemed to work out.
Oh, and the guitar string technique (which I saw mentioned somewhere online) bent the legs of the original F103, so it’s probably useless now.
This is only my second time ever doing surface mount soldering, and the pitch was much finer this time. My fine motor skills are pretty average, I think.

