Hi Protoncek,
Great page. According to these photo’s, I should have an ESP-WROOM-32 then, right? All those names and types 
The Bluetooth integration / Missing Integrated Bluetooth or USB Bluetooth Dongle was also my thought at first. I asked this in another topic. But it should not be necessary as balloob mentioned here: link
I do not understand what you mean by: Does your ESP32 module “see” your BT adapter, connected to HA?
The ESP32 Bluetooth Proxy is added as an ESP Home device.
But I also presume the ESP32 Bluetooth Proxy can detect and show Bluetooth Devices in the log without being integrated into Home Assistant.