If you open the driver's door BEFORE the "shutoff" process is complete, the car beeps a warning beep. Not sure why.

As far as the "buzzing in the morning" that is normal also. See the page in the manual that has all the "you may hear this noise but it is not a malfunction" page for a good explanation of some of those noises.

I have resigned myself to the fact that all the mysterious noises the car will make are just a part of the 4th Generation HSD system, and if something is wrong, then the car will not drive. Other than that, I'm getting used to ignoring the noises.