Essentially yes. When air becomes hot it is because it is absorbing energy in the form of heat. The absorbed energy makes the molecules in air move and expand, therefore decreasing the airs density. The opposite is true for cold air. It is more dense because the molecules are closer together and they are closer together because the bonds are absorbing less energy and therefore do not move as much. Hope that helps!

You are exactly right. The molecules in hot air are moving faster than the molecules in cold air.

Because of this, the molecules in hot air tend to be further apart on average, giving hot air a lower density. That means, for the same volume of air, hot air has fewer molecules and so it weighs less. So since cold air weighs more, it sinks, driving the hot air up,

Basically yes. Our atmosphere, like a body of water, is a fluid that has a density that is determined by a balance between gravity and pressure (plus temperature).

Pressure is generally higher the closer you are to the earth's surface. If you put a "bubble" of hot air in a "sea" of cold air, the hot air can have a lower density but the same pressure as the cold air. This bubble then floats up until it reaches a level in the atmosphere where its density matches that of the surrounding air.

Here is a web page that discusses how hot air balloons work which also has some more detail:
hot air balloon