I just got off the phone with Apple support. Long story, but they are going to submit information to the engineers. Whether anything will come of that is unclear, although they said they (support) would be in contact with me.
I would urge everyone here who is having an issue with USB and iOS 7 to contact Apple Support and spell out exactly what is happening, and insist that others are having the same issue and that the engineers should look into the issue (if they haven't already). Give them the model of your car stereo, when it is happening (plugging in, unplugging, turning the stereo on, turning the stereo off, etc), and mention this thread. Hopefully it will help if they see this is not an isolated issue.
For the record, I had *no* issues like this with iOS 6, regardless of what mode the car stereo was in. And still no issues with Bluetooth. Although, as I pointed out, even if I'm connected with Bluetooth, if I'm also plugged in via USB and I unplug the USB cable, Now Playing goes blank.
But yeah...call and don't let them off the hook until they offer to look into it more.