Does cooling on PC affect fps?
Yes it will greatly affect your gaming performance, the reason its overheating is because its extremely loaded in the first place. so the less utilization the cooler it will be. Reseat the CPU cooler and correctly apply the rice sized amount of thermal paste.
Does better CPU cooling increase performance?
A better CPU cooler lets you run faster memory.
Does cooling laptop increase FPS?
Cooling a gaming laptop will help keep temperatures in check, and will technically improve performance, but the reality is that you can't improve your gaming experience by putting a couple of fans under your laptop.