The quick answer to this is that molecule behaviour is more complex than we often think. At least at the lower levels, where we live, there's a lot of mixing in the atmosphere, and molecules are alway banging together. Some, like water, tend to stick to each other. Now to answer your questions in more detail:
1) Yes. Hydrogen is very light. If it doesn't bind to something else, it tends to get kicked out of the atmosphere altogether, never to return. So we are left with the hydrogen that is in heavier molecules like water.
2) Water molecules like to stick together. Loosely speaking, they like to be grouped together in liquid water (or ice) rather than to be individual gas molecules. Inside the water, there are lots of molecules moving around and some molecules get knocked through the surface to become water vapour. But the water molecules also like to go back. When as many are going back as are leaving the water, we say the atmosphere is "saturated".
Water vapour is lighter than air, so if we filled a balloon with water vapour it should float, right? Nope. The water molecules like to stick together. So approximately 98% of the vapour would instantly join to become liquid water and the balloon would collapse heavily onto the floor. For balloons to work, we need to use something like hydrogen or helium that doesn't like to stick to itself.
By the way, splashing of rain drops in the ocean is an extremely unimportant source of water vapour, not a "majority". It speeds evaporation up because the drops now have more surface to escape from, but evaporation still takes place rapidly where there is no rain.
3) The lower atmosphere is simply very well mixed. If we could move the Earth into deep space so the sun's heat didn't mix the atmosphere, and wait a few billion years for the Earth to cool down, then the gases would probably form into nice layers according to their weight. (Except it would also be so cold they would all turn into liquids and ices!)
Have you ever made a salad dressing with oil and water in a jar? Left quietly they separate into layers, but if you shake the jar then they are all mixed together. The atmosphere is very well "shaken", at least for the lowest 15-20km. That is why industrial air pollution can be found in the Arctic and on the highest mountain tops.