The phone certainly plays a part in it, I had the same experience. My last phone was fine with a cheap cable in the same car. Soon as I got the S4...horrible, awful noises.
It's a ground loop problem, I had the same thing. Go buy this and use it as your audio cable, fixed it for me:
Amazon.com: Kensington Noise Reducing Car Audio AUX Cable for MP3 or SmartPhone, including iPhone 4S: MP3 Players & Accessories
Glad to see I'm not the only one, this has made me late to work 3 times in the last month. Notification LED is on or blinking, but phone will not respond to any input.
This has happened to me about 4 or 5 times in the month I've had the phone. Worst is when it does it overnight and I don't get my alarm. You have to pull the battery to reset it. Very frustrating.