No it depends on power, but the power dissipated by the cable, not the power through the cable. The power dissipated is i^2 * r, where r is the resistance of the cable and i, crucially, is the current through the cable which depends on the power it's supplying (which with a resistive load, in this case, is i * 12v).
Yes, but it seems the connectors, not the entire cable are too high resistance.
Using a larger diameter wire would drop resistance in the cable, but if it has to go through the same connector, it will likely still get hot.
Also, NVDA might be telling the truth about poorly seated connectors, that could raise resistance and heat significantly. That could also be handwaving away a business decision to move forward with a design with too little margin.
