A long time's passed since my last update...
The board is basically working (if you solder the diodes "backwards") as far as generating voltages goes, but soldering the connector is still really hard. The one connector I may have managed to stick to the board, is at least partially ruined - the lid won't stay shut anymore - and even if I keep it shut I don't know if it's really soldered properly. Anyway, after hooking everything up to the screen etc., it didn't work.
I've ordered some spare parts for basically everything including the screen itself, as I don't know if I fried the old screen. It seems this time I got a screen with all 39 pins wired, while the old one doesn't actually use all of them. (this is something Sprite_tm ran into in his project.) So the old screen should be easier to handle.
I've also ordered some FPC cables so I can attach them between my board and a spare connector, and test the pins with a continuity tester, but they haven't arrived yet.
Finally, I've fried my Teensy, so now I've switched to ESP-12Es. I've got MicroPython running and it's nice to know that if I manage to fry them, at least they were cheap.