First off, the size wire needed is relative to the amount of power required. Even 4AWG can be wat too small for a run the the hatch of a Supra for large loads.
Assuming that you aren't drawing more than 100A, you SHOULD be able to get away with 4AWG, though I might recommend 2 AWG for that kind of load, or even 1/0. 60-80A is a good place for 4AWG, and less than 60A, 8AWG should be okay. (Though my preference would be to use 4AWG or better for anything over a 40A draw.
Note that this is power draw, not amplifier wattage. Class A amplifiers draw a lot more current for output power, and class D amps draw a lot less. Since most amps are class A/B, 50A draw is usually about 180W in actual power. (I'd rather not get into RMS vs Max vs PMPO. I refer to what's usually referred to as RMS power)
First off, there's a fairly easy way to check your power cables for noise. Use a set of PC speakers and a 10uF capacitor. Connect the 10uF capacitor to the power line, and connect the other end of it to the tip of the PC speaker plug. Ground the shoulder of the plug. Turn on the PC speakers and start the car. If the lines are noisy, the speakers will play the noise. The capacitor will filter out the DC, leaving only audible noise going to the PC speakers amplifier. I've managed to locate and fix a few noise problems with this method.
For the signal cables, the first step is to disconnect the head unit. If that doesn't eliminate the problem, then disconnect the signal cables from the amplifier. If the power is clean, and you're still getting noise from the speakers with the signal cables disconnected from the amplifier, then your amplifier has got an internal fault.