Pico's ESP8266 weirdnesses (Module not found)

The new one stops somewhere in the middle when pasting larger code pieces

Yes, hopefully that'll be fixed soon with the new USB changes - I'm still fighting trying to reliably fix that exact same problem. On the Web IDE it should work though because it delays a bit more than screen does.

can we expect to have a functionality to send Checksummed code

I guess it's possible - it'd have to be some strange extension to the console.

However it shouldn't be needed. My understanding is that USB already has checksumming built in, so you know the data you get is valid and arrived. The fact that data is getting lost is a bug in the USB code, which needs to be fixed.