Web IDE (all version through the website and downloaded) does not show any devices after selecting Web Bluetooth
I'm pretty sure the downloaded Native Web IDE shouldn't offer you Web Bluetooth as an option at all - make sure you're using this one: espruino.com/Web+IDE#as-a-native-application
I also downloaded the ST driver
That's only needed for STM32-based devices connected via USB
I get "Try again, and make sure your device is discoverable"
Are you sure Puck.js is not still connected to some other device? Puck.js can only connect to one computer at a time.
Also, have you updated Puck.js's firmware? Newer firmwares advertise more often so are more reliable to connect to. espruino.com/Puck.js#firmware-updates
I am starting to think that puck-js is not really compatible with 8.1
It is - however it's possible that your PC's built in bluetooth adaptor or driver doesn't correctly support Bluetooth LE devices. Support seems to have been a bit flakey until recently.
I am writing a C# code
You could try using this as a base: github.com/espruino/winnus
It's C++, but should translate quite easily to C#
It's what I use for the Native Web IDE, which should definitely work (users have had it working fine with Windows 8.1)