The radio's pointing to a blown fuse too if you ask me. It sounds like the ignition live feed isn't getting through, and so the radio's behaving as it should if the ignition is turned off. If I'm right, you'll be having to switch the radio on and off manually at the start of each journey. It should just come on with the ignition.
If you have the time and inclination, check all of the fuses in the fuse box, including ones that don't have any obvious connection to the bits that don't work. Sometimes a blown fuse on one circuit can have a knock-on effect on others. Ultimately, as Otto says, the car's under warranty, so you always have the option of just letting the dealer sort it.
You'll probably find that the CD player is standard on the higher models (probably Dynamique and upwards.) The Authentique is one of the entry level ones, so they do cut corners on spec
. Though believe it or not, I miss having a cassette player in my car. Audiobooks are much better on cassette, because cassettes 'remember' where they were up to, even if you take them out and listen to something else.
The best approach for adding a CD player may well be to add a CD changer to the existing head unit. This will retain the factory look and keep the display and controls, and may well work out cheaper than buying a half-decent replacement head unit.