At any temperature $T$ the equipartition theorem tells us that a molecule in an ideal gas will have a kinetic energy of order $\tfrac{3}{2}kT$. Since kinetic energy is related to velocity by $E = \tfrac{1}{2}mv^2$ we can use this to estimate the velocity of gas molecules at room temperature, and for argon and nitrogen this works out to around 400 to 500 m/s.
So the gas molecules in your container are all whizzing around at hundreds of metres per second, and unsurprisingly this means they mix with each other. The speeds are so high that the different weights of the two types of molecule has essentially no effect in separating them.
Gravity does have an effect, and this is described by the barometric equation:
$$ \frac{\rho}{\rho_0} = \exp\left( \frac{-gM(h - h_0)}{RT} \right) $$
This relates the density $\rho$ to height $h$. $g$ is the gravitational acceleration and $M$ is the molar mass of the gas. Let's take your one litre container, in which case the height of the container is around 10cm, and the molar mass of nitrogen is 0.028kg. If we feed these values into the barometric equation we find the density ratio between the top and bottom of the container at room temperature is:
$$ \frac{\rho}{\rho_0} \approx 0.999989 $$
This is for nitrogen. The molar mass of argon is larger at 0.04kg, but this only reduces the ratio $\rho/\rho_0$ to 0.999984. So although in principle the argon to nitrogen ratio changes slightly from the bottom to the top of you container the change is vanishingly small. In practice the gas composition remains uniform.
If you make your container very large, e.g. about the thickness of Earth's atmosphere, then the change in composition does become measurably large, which is why the composition of Earth's atmosphere does change with height.
The second part of your question is a bit different since it starts with separated gases and asks how quickly they would mix. Given the gas molecules are travelling so fast you might think the mixing would be extremely rapid, but in fact this isn't the case. The trouble is that gas molecules collide with each other and ricochet back in random directions. The average distance a gas molecule travels before hitting another molecule is called the mean free path, and at room temperature and pressure the mean free path is around 0.1 microns.
So even though a nitrogen molecule is moving at about 500 m/s it's moving at random not in a straight line. If you start with two gases separated into different layers then the mixing is surprisingly slow. In fact as discussed in the question Why is it difficult to mix helium and nitrogen gases? it can be so slow that it takes days or weeks to happen.
Assuming you aren't stirring the mixture in any way the mixing takes place by diffusion, and will be described by the diffusion equation.
When two identical gases mix, the state is generally indistinguisable from the previous state. If one molecule from the left of the partition changes places with a molecule from the right of the partition, does the mixture actually look any different? If the left and right molecules are identical, you would never know which ones started where.
So entropy doesn't change. It doesn't change because you can't tell the two states apart.
Contrast this with the mixing between two different gases. When a partition is removed and an argon and several helium molecules switch places in space due to collisions, now you can actually see a difference. The state is identifiable and different from the previous state. This means the entropy has changed.
Best Answer
It's true. Special equipment and a long time is required to mix helium and nitrogen. According to one study, a mixture of 2.7% He, 93.3% N$_2$ at 800 p.s.i.g. required a special cradle to repeatedly upend the cylinder, and 20.5 hours to reach equilibrated gas, which then remained mixed: http://doi.org/10.1021/je60005a002. The helium repeatedly slid from one end of the cylinder to the other. The authors overcame this difficulty by devising a mixing mechanism internal to the cylinders.
The molecular weight of helium is 4.02, and density is .1786 kg/m^3 at standard temperature and pressure. For nitrogen, molecular weight is 28.02, and density is 1.2506 kg/m^3. Here's a table of molecular weight and density for various gases: http://www.engineeringtoolbox.com/gas-density-d_158.html.
Helium doesn't mix easily with nitrogen because of the great difference in their densities. But once mixed, the gas molecules are close together and they move around quite a bit with kinetic energy so they stay mixed and don't separate out into layers.