In my state it is against the law to hold your phone while you’re driving so I keep it connected by bluetooth for calls. It’s not a problem when I’m listening to music or podcasts which I frequently do. It’s only an issue when I’m not listening to anything. I’ve looked into updating the car’s software but it’s already on the most current version.
A head unit is the actual radio hardware in a car. Some of them can be replaced with after market hardware. Yes, when my phone is connected to the radio by Bluetooth, I have to keep the unit on and set to the Bluetooth input in order to hear notification sounds. If I have it off, or set to another input I won’t hear any notifications that come in.