How can I make a message appears on my PC when my phone is ringing? When I'm at work, I usually have a headset on my head and I am listening to music. My phone is always on vibrate to respect co-workers.  However, I don't keep the phone in my pocket because of its size, so I don't feel it vibrating.  I miss a lot of calls.

I was wondering if it's possible to forward "phone is ringing" notifications to the PC, ideally via a Bluetooth connection.  I would not mind installing a 3rd-party app on the phone or software on the PC.
Currently Android doesn't have such functionality built-in.

However, there are 3rd-party apps that can do what you ask. Desktop Notifications is one such app.  It requires installation of the app on the phone, as well as a browser plugin on your PC (Chrome and Firefox are currently supported.)  Once both pieces are installed and configured, any notifications (including incoming calls) will pop up on your desktop.  The only limitation is that this is done via data connection, not Bluetooth, so both your phone and your PC must be connected to the Internet.

On your desktop you may want to enable background apps to run when Chrome is closed to ensure you continue to receive notifications.  This option can be found in Chrome advanced settings -> "Continue running background apps when Google Chrome is closed."