Only the OEM navigation, or the dealer can adjust the time the lights stay on. The car should not be beeping if the lights are on auto when you turn it off, they have a delay before they turn off but they do go off.
I never leave mine on auto either

There a light sensor that detects the amount of light it's receiving. During rainy or foggy "daylight" conditions, your best bet is to keep it on manually.

YES, even if you headlights are on, it will turn itself off automatically. I would know because I have my headlights on all the time - day or night. It just takes longer than the "auto" setting.

The is the law is most states. Some states even got signs indicating that.

i wish the car had daylight running lights like the corvettes..

Why?!? Nobody ever want's them, if you want your lights to be on there is this funny little knob on the left of the wheel you can twist ;)