Serial Keyboard / Mouse to Websockets/Xdotool Keyboard & Mouse events :)

Also I was wondering how would appear a pico acting as a hid device + an HC-05

Yes, the Pico could be USB HID, but HC-05 is just standard Bluetooth serial

I have a faulty pico that doesn't seems to respond to anything ( and that has red & green leds lit but not pulsating when connected to USB while its button is pressed )

Could start a new thread for that? Are the LEDs lit brightly? Can you make sure the USB contacts are properly clean? Either use contact cleaner, or even a pencil eraser will often work.

Is it possible you powered the Pico via the USB connector from voltage significantly over 5V? Sometimes that can cause the USB voltage detection circuitry to blow up - but even if that's dead, you could still program it over serial.