No, you don't have to use the wood ash if it is not available. For us, we heat with a wood-burning fireplace in the winter so have a good, free supply. The rest we have to buy. The other 3 together would work just as well.
Just painting the coop thoroughly will help to prevent the mites as they will have no place to hide in the wood. A dust bath is a good idea, too. I make mine with 1 part sand, 1 part peat moss, 1 part wood ash, and 1 part diatomaceous earth (food grade, NOT swimming pool).