They tend to "accumulate" near the ground because that's where the leaks are happening, near the ground. Stop the leak (which otherwise continues to refresh the dense part of the gradient) and give it time, and the gases will fill whatever space is available, homogeneously. The molecular weight of gases is basically trivial compared with their kinetic energies. There are noticeable "permanent gradients" influenced by molecular weights when you compare gas mixtures at sea level vs the edge of our atmosphere, but on the scales we experience, no.
No, absolutely not.
Propane, butane, propene and other gases, believe it or not, do accumulate to the ground because they are "heavier than air".
It is absolutely a given that LPG will accumulate on the "bottom" of the room or the container.
If you have a bottle and you fill it with LPG and air, the LPG will fall to the ground.
I am sure that you will be able to verify that with a quick search on the internet.
You can make an experiment: just close the windows in your basement, then open an LPG cylinder on the last steps of the staircase going to the basement, then remain on the upper floor.
You will feel the odour.
Now you light a lighter, while remaining on the upper floor: no explosion.
Then you throw the lighter on the basement (or anything incendiary) the your house will explode in a very spectacular way ;-)
The cylinder was almost on the ground floor, but the gas collected on the basement and that is why your house blew up.
Just ask anybody working with LPG or having a car going with LPG (5,6% of circulating cars in Italy, mine is one).
An LPG-car owner knows that he is forbidden to park on certain floors of a garage (garages which have a particular air circulation due to chemneys are exempted until -2 if memory serves). In normal garages an LPG car cannot be parked in an underground stage, whereas a methane car (CNG) can. Special parking places are reserved for LPG cars in parking lots with subterranean "grounds".
That is because CNG has a very different behaviour than LPG.
If you do a bit of searching around you will find this and I tell you this is physics, or if you prefer "real world physics".
For instance you can search for parking norms for LPG vehicles as opposed to parking norms for CNG vehicles.
Or just search for "propane heavier than air" etc. (or denser than air which is probably more correct, but people say "heavier").
EDIT: the different behaviour of LPG and natural gas is know to homebrewers who use LPG "turky friers" for their brewing needs. LPG accumulates on the ground where it can explode quite dramatically.