Air does fall under gravity, as other comments. The other side of the coin, the reason that there is not just a few feet of super dense oxygen and nitrogen coating the surface with the vacuum of space starting just above, is due to thermal motion. At surface temperatures air molecules are bouncing around at average speeds of several hundred mph. Without gravity they would fly off into space. So the two factors balance out with the atmosphere stretching out, getting thinner and thinner, to an arbitrary limit usually taken to be 100km above sea level.
It does fall to the ground. There’s more air closer to the surface of the earth. Think about a pile of sand. The closer you are to the ground, the more sand (air) there is.
For your second question: Fluids exert pressure. Pressures create force (buoyant force) on surface of objects. If the force is greater than the weight of the object then the object will float or rise to the top of the fluid.
The denser the fluid, the greater the pressure. The lighter the object, the lesser the weight.
Air falls due to gravity. Like water does. Water in the oceans does not all fall to the bottom of it, there’s a large pile of it as water has nowhere to go except on top of other water. Same with air. That’s why we have pressure (all that air above us), and that’s why pressure is less on Mt. Everest than at sea level, as the pile of air is not as high from that altitude.
If you want some practical examples of what the comments are mentioning, there’s a common school experiment involving paper bags attached to a pole, like a beam balance. You can use CO2 from dry ice, or combining vinegar and baking soda, or however they did it, and pour it into one of the bags. CO2, being heavier than most other gasses in air, will make the bag sink.
You can also float a paper/foil boat on sulfur hexafluoride because it is so heavy.
As others have pointed out, air *does* fall. Air is a fluid. Denser fluids fall and displace less dense ones. The situation is easier to visualise in liquids, which are the other main type of fluid. A random but interesting example of this is found in the “seven-layer” tea that can be found in Bangladesh. https://en.wikipedia.org/wiki/Seven_Color_Tea
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