Can I make calls via my android phone over my desktop (and the bluetooth headset attached to the latter)?

Solution 1:

It may work with scrcpy over TCP/IP (Display and control your Android device) and the tool sndcpy (Android audio forwarding (scrcpy, but for audio)). I have not yet time to test it myself, though.
A discussion regarding forwarding audio and scrcpy you can find on the Github page of scrcpy, another suggestion was to use scrcpy and use bluetooth for audio.

Solution 2:

It is working for me with Kubuntu 20.04 and I'm using it exactly as you describe! All audio is routed through the Kubuntu laptop.

Android Phone <=two-way audio=> Kubuntu Laptop <=two-way audio=> Bluetooth Headset

You need two Bluetooth adapters in the PC because only single audio BT profile can be used on each device.

The Android phone is connected to the first BT adapter which is embedded in my laptop using Headset audio gateway HSP/HFP profile.

The headset should be connected to the second BT adapter (USB dongle) using Headset Head unit (HSP/HFP) profile.

These profiles can change automatically when a call is established and sometimes get stuck in the wrong profile and needs to be set manually.

Also you should see where the default pulseaudio sink and source are and try to find the correct configuration in the advanced settings of Volume control.

What is working: Phone calls, playing multimedia files in the phone, whatsapp calls (I only hear choppy sound - the problem is between phone and laptop, the remote part hears my voice OK), all audio playing in the laptop.

With "KDE Connect" it is possible receive the notifications connecting by WiFi.

The headset button stops/starts multimedia player on laptop and it is not working / not yet configured to answer the call.