If deep litter is dusty, it's because it's too dry. Dust can be from the protective coating on the blood feathers, bits of litter, dry broken down manure, food that's gotten spilled or even powder that some people add to their coops. Having a minimal amount of moisture will keep the litter...
Just wood shavings, chicken poop and some moisture will compost. You don't need anything else. It's more about the amounts of each, just like in a compost pile. Some people use other litter choices, I normally use wood shavings.
If you read anything about composting, it will help explain...
Litter can start composting in a coop with a floor. Just like garden refuse can compost in a compost bin that has a floor. It's about what ingredients are in the mix and the moisture levels in the mix.
Is it a little easier and faster if you have them both on dirt? Sure. Partly it's because...