Not sure how I missed that in your original post, but I see it now.
Noise can get into the radio in a number of ways, through the power feed, through the antenna, or through any other wiring (speaker, accessories, microphone, etc).
The fact that it does it with or without the car running helps a bit, but doesn't narrow it down. Likely it's not ignition system noise, although there are many other items in the car that stay energized with the key off.
Feeding directly off the battery can solve some issues, but you have already done that. You might want to try making sure the negative lead is not connected to the negative post on the battery. Connect the negative lead for the radio power to a body ground, preferably one near the radio itself.
Make sure things like cell phone chargers are unplugged, some of them can be pretty noisy.
If that does't solve it, try pulling the fuses one at a time and see if the noise disappears.