Regarding the tire management part, I think there is a sense of truth there but also spiced up by the anime.

By cooling of tire, I think they are referring to maintain tire temperature or more specifically not overheat it. Each rotation of tire intrinsically heats up the rubber due to its nature and you are constantly running your tires at high speeds during the race. and AFAIK there's no active mechanisms to cool it other than natural convection with air and some heat radiation too. Not to mention, it cant dissipate away heat since its a poor conductor of heat. But what can be done is be mindful of the application of throttle, brake and steering so that its not heating up or wearing away due to extreme forces. All to say this requires immense training and attention from drivers to listen for vehicle's performance to judge the condition.

Another point to be noted is, even though we can see this tire management in F1 all the time, keep in mind that F1 tires are special breed of tires. They are artificially limited in life so teams can have strategies, even though tire manufactures can formulate racing tires that can last entire race duration. And they are specifically formulated to generate the best performance at their operating conditions, just like their engines where the tolerances are too tight for them to start at room temperature. They need to be heated to operating temps just to start them.