I am using a different dock (the iBolt xProDock for the Droid DNA that I got from Verizon, which is the same as the iBolt one for all HTC phones, but does not include the cable and usually is $10 less) and have had issues with getting it to auto launch the HTC Car app and the iBolt Car dock app which I am using now. I have been using the app called "Car Mode Control" by IG88 Apps to force it to go into Car Mode. This app has not been updated for a while, but seems to work for me unlike IG88 Apps actual Car Dock app. Car Mode Control allows you to set the default app to launch when going into Car Mode. I actually like the iBolt Car dock app for the most part and it is currently free.
Most of the time when I unplug from the dock the Car Mode quits, but not every time. Then I have to go to the notification bar and select the item to exit Car Mode. Not sure why it is sensing on unplug, but not plug in?
Be aware that I have heard when the HTC One gets upgraded to 4.2.2 that Car Mode may stop working at all since Google removed this feature from android 4.2.