Android - Can I get my car's head-unit to play audio, via USB, from my Galaxy S3?
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.
As both eldarerathis and Paul Redert have mentioned, it seems that the USB port on your Juke is designed to support two types of USB connection: 1) mass storage mode, where it reads an attached USB peripheral as a logical drive and searches for compatible media files, and 2) iDevice Accessory mode, where it attempts to communicate with and Apple device over a proprietary protocol.
CNET has an article from 2010 describing the different ways Android phones can connect to a vehicle (as of 2010, of course). While most of it will tell you what you already know, I thought their compromise solution might be applicable:
...during our testing of the 2010 Acura TSX V-6, we were able to pair our Motorola Droid via Bluetooth for phone calls and Pandora Radio while using the USB port for charging. When we wanted to listen to locally stored audio, we simply mounted the USB connection and used the AcuraLink receiver to browse our folders. Likewise, a user could do the same thing with a vehicle that has an auxiliary input and a USB connection, but no Bluetooth.
It's sounding like mass storage mode might not be an option for your SGSIII, but using the Juke's USB connection for power to offset the use of Bluetooth for media might be a way to accomplish what you want. If you're worried about forgetting to turn off Bluetooth, you could look at ways to instruct the phone to remember for you. My personal favorite is Tasker, and setting up simple profiles for turning on and off Bluetooth under specific conditions could be as simple as two one-line tasks.