Suppose the pressure at the Earth's surface is $P$.
Consider an air column of cross-sectional area $A$.
The upward force on the column is $F_{\text{up}}=PA$.
Denote the weight of the column as $W$.
By definition of "weight", the downward force on the column is $F_{\text{down}}=W$.
Suppose the pressure is too low, such that $F_{\text{up}}<F_{\text{down}}.$
The column of air will then fall downward.
As it does, it more air molecules are arriving at the surface of the Earth, increasing the density of air and therefore also increasing the pressure.
Since the pressure increases, so does $F_{\text{up}}$.
This will continue until $F_{\text{up}} = F_{\text{down}}$, at which time the system is in equilibrium and stays the same.
In other words, the pressure is such as to balance the weight of the column because that's the only situation which won't immediately change.
I) But haven't masses in vacuum not the same attraction and speed.
No. Their weights are different, so they are not "attracted" / pulled in by gravity equally.
Think of this: If you find 100 heavy perfectly round stones, and you put 5 plastic balls full of air with exactly the same size in the basket with them, what will then happen when you shake them a bit? Will the lighter plastic balls fall to the bottom or "float" to the top?
They will float to the top.
The point simply is that it is easier for helium atoms to move up than for air molecules. If you shake the basket violently, the stones might jump a bit while the plastic balls can jump much higher. So on average, the helium atoms will move much higher upwards, and as soon as they do that, some oxygen molecules will take their previous location. Now they have a new location higher up, and the same happens.
Overall this causes the effect of buoyancy, sometimes called updrift, which is the force that this lighter material is pushed up with. And this upwards force is exactly the same as the force, with which the heavier materials pulls downwards - in other words, the lighter material is pushed up with the weight of the displaced heavier material, which now pushes to come back in place.
This was Archimedes' discovery.
Now to your other sub-questions:
Can be said that for airmolecules the atmosphere is a vacuum?
Well, no, a vacuum is a vacuum. If there are molecules present, it isn't vacuum, and the atmosphere isn't a vacuum.
So all together helium should have the same attraction to earth as the other airmolecules?
No, their "attraction" to Earth are different, because that "attraction" must be weight. And the helium atoms weight is lower.
II) Because helium atoms are much lighter, perhaps they could have a higher speed than fe O2 or N2?
Mass (or weight) doesn't influence possible speed. It only influences how hard it is to make them reach the speed.
Ok, but those helium atoms are in a balloon so they pushes at all sides of the balloon equal so the balloon shouldn't move at all?
If only the balloon with helium was present, and no gravity or outside atmosphere, then you are completely correct. The inside pressure cannot make the balloon move. But with gravity present, the whole thing is pulled downwards, and with the atmosphere present, there is a buoyancy force upwards as discussed above. Which-ever of these forces is greater, makes the balloon move.
III) When a balloon starts ascending from the ground there is more air (pressure) above him than beneath. So the airpressure above him should push the balloon to the ground?
Incorrect. You actually said it yourself just before: Inside the balloon, the pressure equalizes throughout so the push at any point on the balloon is the same. Same goes for this air column: All the air in the column above presses down, but the tiny bit of air below pushes up with the same force to balance out the pressure.
Best Answer
Gas molecules move very fast and tend to mix more than they tend to settle due to gravity and density. Similar to what happens inside any bottle of liquor. Alcohol is lighter than water but it doesn't float on top, it stays mixed in. The water and the Alcohol mix naturally in part due to their shape and in larger part, due to their charges. Gases aren't quite the same as what happens with water mixing with alcohol, but the effect is similar. Water tends to mix into the atmosphere more than it wants to float above it and the high speed of atmospheric molecules tends to keep the atmosphere well mixed.
The more important factor regarding water vapor in atmosphere is temperature. Water can both evaporate into air and condense out of it and it tends to do both at the same time to some degree, leaning towards an equilibrium based on the specific circumstances. As air gets warmer it can hold more water vapor. Surprisingly more. Every 1 degree c, the atmosphere can hold as much as 6% more water vapor, so, ballpark, every 11 or 12 degrees in temperature change doubles the amount of water the atmosphere can hold. A typical summer day, provided it's humid heat, not dry heat, the air you're breathing is over 2% water.
The other thing that happens to warm air is that it's lighter than the cold air above it. This lighter air wants to rise and as it rises it cools, and as it cools it can no longer hold all that water vapor, so the water vapor tends to form tiny drops of water or bits of ice, which begin to fall towards the earth, but quite slowly, and remember, they are falling in an updraft of rising warm air, which creates the effect of clouds appearing to hover in the sky, when it's usually a combination of slowly falling ice crystals (which fall about as fast as very light fluffy feathers), and an updraft.
Pictures of warm air rising added
Water vapor in the atmosphere is transparent. It can only be seen as clouds as it condenses out of the atmosphere. (Same with fog). The mass of the individual molecules isn't very important.
As to the size of the gas molecules, that's not important either, not in a gaseous state. The standard atmospheric formula, P1V1/K1=P2V2/K2 doesn't take into account molecular size. Now for liquid or gas, molecular size matters.
You also seem to have the wrong size for your molecules, maybe you took an individual Nitrogen atom. A water molecule is one of the smaller molecules, because hydrogen is so small and tightly bound to the Oxygen. It's about 2.75 angstroms. A Nitrogen Molecule (N2) is about 3 angstroms.
Water is also polar, which helps it stay liquid at higher temperatures because the lightly negative charge on the O tends to bind with the H molecules on neighboring water molecules. Nitrogen doesn't do that, so it only becomes a liquid at very cold temperatures and despite being a heavier molecule than water (by a fair bit, 28 to 18), liquid nitrogen is about 81% as dense as liquid water. It doesn't fit as tightly together. (see 2 pictures below). The 2nd one doesn't have Nitrogen, but Nitrogen is actually a slightly larger molecule than Oxygen, which as you see, Oxygen is slightly larger than water. Water is both smaller and it fits together more neatly, so it's surprisingly dense compared to what you might expect looking at it's atomic weight.
All that said, to your question, are lighter gas molecules affected by gravity, The answer, I believe is yes, but it's a very very minor factor. Wind, mixing and chemical interactions like evaporating and condensing are larger factors. Ozone, for example is quite a heavy molecule, but it's formed and broken down high in the atmosphere long before it can fall to the earth due to gravity. CO2 is a heavy gas but the atmosphere mixes enough that it has no problem providing plants high on mountains all the CO2 they need.