Does the propeller to old planes help to cool down the plane engine?

Simply having the engine exposed to the airflow is sufficient to cool an aircraft engine. Besides, the part (root) of the blade close to the hub really doesn't move much air anyway.

The fact of the matter is that you don't want to excessively cool an air-cooled engine anyway or you reduce thermal efficiency in the engine. That's why most have cowl flaps and some even have diffusers in the cowl opening.

Many famous fighters with propellers engines, had engine cooled by glycol and oil -

Propeller did not provide any cooling to engine -

Examples are P-51, Spitfire and Messerchmidt Bf.109 -

Air cooled engines are really not cooled by propellers -

But actually by the relative wind (airspeed) -

Examples of air cooled are Hellcat F6F, Thunderbolt P-47, Corsair F4U

Not really.

The engines are mostly behind the lower part of the blades, near the center of the prop. The closer you get to the center of the prop, the less air gets blown. It's mainly the tips that do all the air blowing.

Not really. From a technical stand point, propellers are actually wings that rotate around a central axis, providing thrust, if they're mounted on an airplanes' center line, or thrust and lift as on a helicopter.

NO. The propeller is there to cool down the PILOT. If it stops turning, then the pilot really starts to sweat.

yes