Pico's ESP8266 weirdnesses (Module not found)

100uF would be great (I used 47uF), but I'd solder it right onto the top of the ESP01 module. If it goes into breadboard first it might not be as useful.

I'm very surprised about the Pico reset, but if it can do that then it's not a big surprise the USB connection drops out occasionally. Could it be the USB extension lead you're using has quite a high resistance and can't supply the power required? Just a thought...

The shims are not yet available directly/from stock somewhere, are they?

Afraid not... I am planning to add them at some point though. You can still order them relatively cheaply from OSHPark by just uploading the design though

a "special mode" if powered by a computer with a potential serial connection in contrast to just being powered by a power source

Yes, it's a bug with the Pico that I'm trying to fix - I think it's mentioned on FAQ/Troubleshooting somewhere. It happens on the original board too, but only if you manage to fill the output buffer. On the Pico it happens straight away. I'm trying to fix it while adding USB HID support, but ST's new USB library is proving difficult to get reliable.