The tiny one that runs to the front of the car tells the amp to turn off when the ignition is off. It’s referred to as the ‘remote’ wire. It will only see a few miliamps, so don’t worry about it’s size.
As beaten up above, the ground wire from the amp should be the same size as the power wire.
In order to draw more than (75 watts / 14 volts / ~85% eff =) 6.3 amps you are either overloading the amp (which as I said can work just fine) or you have a bad connection. A bad connection will cause the available voltage to drop, changing the calculation above. The amp will still try to do the work it’s set up for, but with reduced voltage it requires more current. (V=IR). An undersized ground, or a ground in the wrong location can cause this.
Are the subs single or dual voice coil? How many connections are on each speaker (speaker itself, not subwoofer box)? If you have an ohmmeter, to check for overloading, see if speakers are 4 ohm. If they are DVC they could be set up for 2 ohm (increasing the output of your amp by ~60%).
If you have the power and ground wires sized reasonably, and the connections are all tight you should never draw more than 15 amps with that amplifier!
Another question: Where is the ground wire for the amp connected to the chassis? Is it only the rear light that dims? How about the headlights? It could be that the rear deck light shares a ground point with the amp, and it’s not a good enough connection point.