The battery is NOT part of the charging system. The charging system consist of the alternator, rectifier and regulator. If the stereo is drawing current from the battery while the engine is running then the battery will eventually have a shortened life span because it is not charging as it...
A second battery is only good for extended listening with the engine off. Once the engine is running all the electrical equipment gets it's juice from the charging system and not the battery. When the engine is running the battery becomes part of the load on the charging system just like the...
If you can hear a difference in your stereo between 4awg and 0awg you've got better ears than any dog I know.
This is a perfect case of diminishing returns here. You're spending more for the larger gage cable for, IMO, no sonic improvement.
Be sure to use RCA cables that use "twisted pair" construction. Twisted pair construction has superior noise rejection characteristics that shielding cannot provide.
Simply put an ohm is a unit of resistance (or impedance) to the flow of current. Resistance applies to DC voltages while impedance refers to AC voltages which are constantly changing in amplitude and frequency.
Here's a good source of info on how to track down unwanted noise in your car's system...
Noise Elimination
Pay particular attention to the part that discusses "muting plugs" in section 8.1. I've used them on many occasions to track down noise problems in cars.