atmospheric scienceearthmassmeteorites
The earth presumably loses mass because molecules of the atmosphere disassociate and fly off into space where the solar wind carries them away.
On the other hand the earth gains mass because particles of dust and meteorites strike the earth and accumulate onto it.
Which force is dominant? Does the earth gain mass over time or lose mass?
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.
The aurora are emissions of light caused by the excitation of nitrogen and oxygen mostly by energetic (i.e., ~1 keV to few 10s of keV) electrons coming from the Earth's geomagnetic tail (i.e., anti-sunward direction). There are proton-driven aurora too, but they are fainter and more rarely observed (partly because it takes much stronger geomagnetic storms to produce enough proton precipitation into the atmosphere).
Note that electrons with this range of energies always exist in the solar wind, however the number fluxes are much lower than those that drive the aurora we all know and love. The solar wind plasma tends to have number densities of ~1-10 cm-3 and bulk flow speeds are typically ~300-500 km/s. To put that in perspective, that corresponds to a dynamic pressure of ~0.07-2.0 nPa, or more than 50 trillion times less pressure than one atmosphere at STP.
Suppose the earth wasn't surrounded by a magnetic field. How would the Aurora look like?
So if the Earth were to suddenly have no magnetic field, there would be a diffuse aurora seen (only by special instruments, as I explain below) at almost all latitudes but it would be extremely dim.
Part of the reason for the brilliance of the aurora is due to a sort of magnetic focusing during the precipitation process. So imagine taking a volume of space much larger than the Earth filled with electrons, accelerating them earthward, and then focusing them into an area that's a mere fraction of Earth's total surface area. The combination of the focused area and high fluxes results in a large contrast between the night sky and the aurora (which are actually much dimmer than a full moon).
Would we see a sky full of colored star-like lights, with the color dependent on the energies of the incident particles, the kind of particles, with which particles the incident particles interact with and the altitude at which the interactions take place?
No, not really. It would probably look more like what we see during very strong geomagnetic storms, namely, dim and diffuse light over much of the sky. However, I doubt anyone would see much as the solar wind is incident on Earth from the same direction as sun light (i.e., only daytime auroras on such an Earth).
Mars has aurora like Earth, but it does no longer has an intrinsic magnetic field. There are surface remnant fields and it is about these that we see faint aurora. Yes, Mars does have an atmosphere, but its much more tenuous than Earth's. In fact, a neutral gas surrounding a planetary body is really all that is necessary to produce emission of light from impacting charged particles, i.e., this is basically to what the aurora amounts.
We have also observed aurora on Jupiter, Saturn, Uranus, and I think Neptune. However, it is mostly in the UV light spectrum, not visible that we observe these.
I should point out that while the loss of an intrinsic magnetic field would be very problematic, as I discussed at https://physics.stackexchange.com/a/214509/59023, it would not mean there would be no magnetic field.
There would be two types of ionizing radiation: electromagnetic radiation at UV or higher energies and energetic particles. The net result would be a very active ionosphere like that of Venus. This would create an ionopause and a bow shock similar to what Earth's magnetosphere does (the bow shock, not ionopause), but the bow shock would be much closer.
Let's assume Earth lost all of its atmosphere. Without an intrinsic magnetic field or atmosphere to ionize, the particles would mostly pass by the planet on balistic trajectories producing a wake similar to what is seen in supersonic wind tunnels. However, the solar wind does carry a magnetic field, so the particle trajectories around the blunt object would not be purely hydrodynamic (i.e., the Lorentz force still matters).
Best Answer
We really don't know the figures of mass loss, or mass gain through cosmic dust accumulation.
Mass loss through Hydrogen and Helium
If you take the lower end of the mass accumulation estimation from cosmic dust, that is 5 metric tonnes daily, this results in a figure of 1825 metric tonnes a year, resulting (if you take 50,000 tonnes a year of $\mathrm H_2$ and $\mathrm{He}$ as accurate), in a definite answer that the Earth is losing mass.
A break even figure for mass equilibrium is 137 tonnes of comic dust daily, which is almost midway between the (widely differing) estimates of 5 to 300 tonnes of cosmic dust thought to fall on Earth daily.
Cosmic Dust Estimates
The answer is we really don't know, but as we seem to have more accurate figures on mass loss than mass gain, it seems we can have more confidence that Earth has a net mass loss.