As far as putting dirt around the edges of the wire, save your money and effort. The birds will scratch it all away until they get to the wire on the inside and whatever wants in from the outside won't be detered by a dirt berm.
It would be a good idea to clear some, not all of the vines/greenery away from the area to allow some sun and air in to help with the moisture, but still provide needed shade to the birds. Some lime mixed into the soil will help with the moss. You can just sprinkle powdered lime onto the moss and it will kill it.
As far as using just chicken wire on the door, it's not a good idea. Spend a few extra dollars and get hardware cloth and use poultry staples to secure it. Chicken wire and staple gun staples wont do anything but keep the chickens in. Any predetor that wants in can go through chicken wire and staple gun staples in a matter of a few minutes.
The opening in your roof would be easy to cover with fiberglass panels available at most any home improvement store for less that $20. You can secure the panels with screws and it will allow light in and keep the rain out.
I would replace the rotten wood, definately. If you can, it wouldn't be a bad idea to paint the wood that has mold, after it has been cleaned with a bleach solution and allowed to dry, with Kilz. It will seal in any harmful mold.
Soft wood shavings over dirt make an excellent litter.
You must have plenty of ventilation.
Over all it looks like a decent coop and with a little work it should turn out to be a fine home for your birds.
Welcome to the forum!
-Bantyrooster
