The roof will keep rain from falling directly in.
If water flows downhill into the coop space, then you will probably have a wet mess inside.
But if the water flowing downhill gets diverted around the coop, you will probably be fine. A ridge or a ditch uphill of the coop could help direct water to either side and let it flow around the coop, rather than in.
I might throw some dirt in just because the chickens would enjoy scratching through it, but I don't think it will matter much.
I'm not sure how "too deep" would be an issue, unless water is getting trapped in there. Having the solid part be a bit higher than the surface of the litter can be good, because that will help keep the litter inside. Otherwise, it tends to get thrown out through the wire fencing.
For the actual litter material, I would build it up gradually, using whatever suitable materials are handy. For example, you might start with some wood chips, add vegetable peelings and other compost-type materials as you have them, shred the junk mail (paper) that arrives in your mailbox, add the weeds you pull from your garden or lawn (only if not sprayed), pile in lots of dead leaves in the fall... Each time you dump in more stuff, the chickens will spread it out and mix it into what is already there.
If it ever smells bad, check whether water is flowing in, and add more dry material (because smells usually come from being too wet.)