Does there exist an in-car device that does the following things?

I believe it depends on the head unit in your vehicle. My wife has the stock stereo in her Toyota Avalon and uses it as a speakerphone all the time. She uses a tape adapter to the headphone jack, and when she gets a call, she just turns the radio up real loud until she can hear the ringer or caller. I don't know why it is so much quieter than the music she is playing.

I've tried it in my truck as well, it works well. I think that takes card of your #4.

It sounds like you have a head unit with an 1/8" aux input, is that correct?.

The way I see it, you could get a mounting charger, and a short 1/8" cable to run from your headphone jack to your aux in. It's not one device, but it should work.