Cat Man is right, "coolant" is a misnomer. I remember when coolant had an accurate name. Anti-freeze/Anti-boil. It doesn't really make the engine run cooler, it raises the boiling point.

As the liquid heats, it expands, creating pressure above atmospheric(in a sealed system), which raises the boiling point even higher. The cap allows pressure to build to a point, then release.

Of course, if the system has a leak, external or internal, the cap is rendered pretty useless, and the liquid can boil. Boiling is the enemy, creating air pockets, and very poor heat transfer. Like a chain reaction.

 

"straight water is the most effective coolant to remove heat"
Do you mean to say all of those mechanical engineering classes were just a waste of my time? Density of the medium has no effect? They should quit using liquefied salts/metals for cooling nuclear reactors in favor of straight water?
It is the actual "boiling" of the water that is the problem; i.e. the entrained air doesn't conduct any heat.

 

Your first question
Yes water has nothing added to help block heat transfer. Density retains heat and harder to reject. Steel vs Aluminium. Heat both to 100*F and see which one cools faster. Density.
Second
Air is an insulator. Why do house's have double pain windows? No, air does not conduct heat, it insulates it. It's the job of a liquid or solid to do it.
Having air in the system doesn't let coolant touch the metal surface to do it's job. Having the system free of air is the first thing you do. Second is to move the boiling point above atmospheric (15 psi at sea level). That's why radiator caps have a pressure rating.
When I ran stock cars, it was the norm to run 23 lb caps so the engine didn't over heat.
If you want I can bring up stuff from my Caterpillar Marine PAR training book on this. Yes I am still certified from Cat.
That's where the "Cat Man" comes from.

 

Density absorbs more heat; then it goes to the radiator which is correctly sized to exchange the heat at the optimum rate (calculated for a fluid made of 50/50 water and DexCool).