Can I get my car's head-unit to play audio, via USB, from my Galaxy S3? I've just moved from iOS to Android, purchasing the new Galaxy S3. Lovely phone; I'm impressed thus far.

The only thing that I'm struggling with is playing audio, via USB, to my car's head-unit.

I've got a 2012 Nissan Juke (upgraded Nav head unit with USB port in the center console) which worked fairly seamlessly with my iPhone 4. I'd plug the USB cable in, and my music would start playing through the car's speakers - with names/album/length etc. displayed on the head-unit's display. This worked with music I'd loaded on to the phone and the Spotify app.

When I plug my S3 in, the phone detects the connection and asks what mode I want to use - Media mode, or Camera mode. Regardless of which mode I use, the head-unit is unable to play either stored or Spotify music.

I've read up a little, and I'm wondering if this is to do with the S3's inability to render itself in Mass Storage mode.

Does anyone have any suggestions for a workaround, or an app which might resolve the problem? I can play audio via Bluetooth, but its interface isn't as seamless and I will forget to turn Bluetooth on/off; don't want to drain the battery excessively.
Yes and no. The fact that your Galaxy S3 does not support mass storage mode means that you cannot play audio files stored on the S3 with the radio, like you could with a USB stick. (If your car's radio supports mass storage, that is.) 

However, the other type of USB connection, allowing your iPhone to be controlled by your car's radio is an Apple Accessory connection. This is a protocol for communication between Apple devices such as the iPod and iPhone and accessories like your car's radio. Android currently does not support the Apple Accessory connection, but has it's own protocol for communication with USB accessories. It might be technically possible to create a solution for Android to allow it to speak with Apple Accessories, but currently (as far as I'm aware) no such solution exists.