Sorry, it is wrong to say there is no air movement. Crudely, whilst the movement of bulk air may cease so it feels as if the air is still, the molecules of various chemicals such as CO2, O2, N2 and others which make up air are still moving. It is this movement of individual molecules which cause diffusion. (You may not realise it but you do feel these individual molecules, not individually but in bulk as pressure on your skin.)
Please think of air as a vast number of molecules which are moving randomly in all directions. Now take a container with an internal dividing wall. On one side we have molecules of O2 & N2 (that is air) and on the other side we have molecules of CO2. If we remove the wall (without disturbing things), those molecules of CO2 which were moving toward the now removed wall will continue onwards into the other half of the container. And like wise, those molecules of N2 & O2 moving towards the wall in the other half will pass into the first half. I hope you can accept that eventually the molecules will mix and the concentrations of CO2, O2 & N2 will be the same across the whole of the container. It is (almost) impossible to reverse this process. So one does not get CO2 concentrating out of air, rather one gets CO2 diluting into air.
If you want to get a visual appreciation of diffusion, then watch
(though convection currents similar to 'wind' are also at work).
Yes some rocks can create CO2 under the right conditions (including limestone). And rocks like coal can have trapped CO2 from when the coal was formed from rotting vegetation (as well as gases like CO - carbon monoxide - white damp). Which is why disused coal mines are very dangerous / suicidal to enter. Also it is possible that rotting wood and other materials taken into a mine can create CO2.
CO2 dissolves in water (just like sugar). The more pressure of CO2 in the air above the water, the more CO2 dissolves into the water. I am going to ignore the complex chemical reactions which can take place. But I will point out that if the water is transported to a different place (i.e. through fissures in limestone rock into a cave passage with ordinary air), then because the pressure of CO2 above the water is lower, the CO2 will come back out. Hence the presence of higher levels of CO2. Normally the CO2 will diffuse into the air (or indeed be blown around such as by a strong stream) and not be noticeable. But some times it takes a while to diffuse and is noticed.
As has been mentioned one can also get oxygen depletion in mines. Usually CO2 levels do not significantly rise so what one suffers is lack of oxygen not elevated CO2. In fact the science of levels of O2, N2, CO2 and other gases such as CO and CH4 in mines is incredibly complex and beyond my level of competence. You also need to understand the instruments which measure gases in air as well. Some misleadingly just measure the oxygen level and for values below 21%, assume the difference is due to CO2 and report it as CO2, ignoring the potential for oxygen depletion.
By the way, so far I only know one person who died in a mine due to bad air (the generic term) so please don't become the second person. I won't say good luck, rather I will say get educated.