Great question. The answer is that both are important in different ways. The air pressure depends both on the temperature (or kinetic energy of the air molecules) and on your altitude (or the weight of the air above you), so I would say that air pressure is caused by both.

Here's how it works. Think of the atmosphere as giant mob of people trying to rush the stage at a concert. Everyone wants to get to the stage, but since there are so many people and because everyone is pushing and shoving they can't all get there. Still there will be a lot of people close the stage and fewer far away. Our atmosphere is just like this, gravity is pulling all of the air molecules towards the surface of the Earth but collisions between the molecules keep them from getting too bunched up near the surface of the Earth. This explains why it's harder to breath at higher altitudes.

As a simplified general model, we treat pressure as the effect of molecules colliding against a surface. (The kinetic energy of the collision creates a force against a surface, and given the number of collisions per time per unit area, you can get the pressure by dividing the force by the area.

Yes! It is the weight of the air above you being pulled towards the earth due to gravity. It is why the pressure goes down as you climb a mountain. It is the same principle that explains the high pressures experienced underwater (water ways a lot more than air.) You suggested that it has to due with kinetic energy, which is not such a bad idea. The average kinetic energy of molecules within a material is actually what defines temperature! Of course, hot air and cold air can have different densities, and thus different weights, but it is the weight that we experience. For instance, if you were to take the exact atmosphere (same amount of air, same temperature, etc.) and put it on the moon, the smaller mass of the moon would exert less gravitational force on the air and thus it would weigh less, and we would experience less air pressure.

Both - pressure is a function of the average kinetic energy of the air molecules (i.e. temperature) and the density of said molecules. The weight of the air above you compresses the air to higher density, and thus higher pressure.