Does CO2 form carbonic acid in the atmosphere?
Short answer

It does form carbonic acid, as rain drops are water and carbon dioxide is soluble in water.
But the content of carbonic acid is minimal as you can guess from $\mathrm{pH=5.6}$

Longer answer

Majority of dissolved $\ce{CO2}$ does not form carbonic acid, 99.83% stays as hydrated oxide.

The equilibrium $\ce{CO2}$ concentration in water follows the Henry's law and is proportional to the $\ce{CO2(g)}$ partial pressure.

Therefore, as a drop falls, the partial pressure of $\ce{CO2(g)}$ grows together with the atmospheric pressure and  $\ce{CO2}$ solubility in water would increase as well, if temperature did not raise with decreasing altitude, typically $\mathrm{ 0.0065 K/m}$.
The total effect is general decreasing $\ce{CO2}$ solubility when a drop is falling.

Sure, some drops may evaporate during their fall ( seen as precipitation patterns without raining ) and the dissolved $\ce{CO2}$ becomes gas again.

Natural rain has $\mathrm{pH=5.6}$  because of dissolved $\ce{CO2}$ and equilibrium concentration of $\ce{H2CO3}$.

If all  CO2 converted to H2CO3, the natural rain and soda water would be much more acidic, because carbonic acid is quite strong acid with pKa=3.6, instead of formal 6.3 if 100% conversion is supposed.

In carst areas, there is ongoing reversible reaction of dissolving calcium carbonate by surface rain water going down the soil and precipitating it again in caves :

$$\ce{CaCO3(s) + H2O + CO2(aq) &lt;=&gt; Ca(HCO3)2}(aq)$$

The calcium bicarbonate comes also to surface water and makes frequently the majority of the calcium content.( and similarly for magnesium l