Question: Also, would an intercooler help the car run cooler even if I dont have a turbo on it?
"Air into the intake snorkel", and "Intercooler" both seem like references to engine combustion air intake, as opposed to engine cooling. Engine cooling is off-subject from the thread; the issues appear to have become confused.
A intercooler is intended to cool, and make more dense, the air/fuel charge. A denser charge further improves the volumetric efficacy of the engine, netting better engine power.
Greater engine power output will tend to increase thermal loads to your cooling system, but that really is another thread.