Hello Ektar,
Welcome to the Mattress Forum! 
The most “natural” mattress materials would be 100% natural latex (see post #18 here), natural fibers (such as wool, cotton, flax linen, silk etc), rubberized coir, and depending on how you are defining natural, perhaps steel innersprings or microcoils as well. I’m guessing you’ve chosen some sort of plywood such as PureBond so as to be formaldehyde free. While the carpeting you have should provide a bit of airflow under your mattress, mold can grow anywhere the environment is hospitable for growth (proper humidity and temperature, and a food source), even for items that are inherently mold and mildew resistant, such as latex and wool. Normally I’d recommend something like a coir bed rug on top of the plywood deck, which would be 1" thick and promote better airflow that a carpeted layer.
Regarding chemicals, any foamed rubber core includes other chemicals or substances that are necessary to manufacture the foam (such as foaming agents, curing agents, accelerators, gelling agents, antidegradants, sometimes fillers, and others) so the final percentage of rubber in a 100% natural rubber core (outside of any fillers it may include) are generally in the range of about 90% - 95% or occasionally a little higher. But if you’re looking for a foam product that uses the fewest “ingredients” (even if they are rendered inert in the finished product), then a natural latex layer would probably be your best option. You could also consider a natural fiber product.
In terms of “safety”, all the latex you are likely to encounter (whether it’s Dunlop or Talalay and made with natural rubber, synthetic rubber, or a blend of both) will have been tested for harmful substances or VOC’s by either Oeko-Tex or Eco-Institute and certified as being “safe” so safety wouldn’t be an issue for most people regarding any type or blend of latex (see post #2 here for more about certifications)
I would start by looking through the site members listed in post #21 here, many of whom can create custom sized latex products (you’ll probably be able to get the most comfort out of your limited thickness needs with latex). I believe that all of them compete well with the best in the industry in terms of their quality, value, service, and transparency. Off the top of my head from that list, Mattresses.net, SleepEZ, Quality Sleep Shop, Rocky Mountain Mattress, CozyPure, and FloBeds can create custom sizes, but I don’t know of the timeframe involved. Other site members may also be able to do custom sizes, but you’d have to phone them directly. You could also create your own product, using the resources of the different vendors in the component post here. A third option would be to stack two 2" toppers together, and there are some resources for toppers listed in post #2 here. However, you’d have to check on the ability of these businesses to customize the size of their toppers.
If you provide your zip code, I can see if I am aware of any mattress manufacturers nearby who also might be able to assist you.
Phoenix