The problem with the plywood is that it can get slick from the poop, making it hard for the chickens to walk safely. This is a really big problem when you have little ones that are still developing, but it can cause problems for the big guys too if their legs slip out from under them.
People use all kinds of different bedding in the coop. Some say straw/hay is terrific and they swear by it. For small growing chicks, straw gets too slick with poop for them to walk on. Other people prefer pine pellets or pine shavings (usually found in the horse stall bedding section of the feed store). Some people use leaves from their own trees, although I am not sure how much the leaves actually absorb. I've seen some people say that a deep layer of sand works well for them, then they periodically scoop poop and add more sand as needed.
Me, I like the pine pellets or the pine wood shavings. personally. I even use pine pellets in my kitty litter boxes. You can scoop poop but leave the pellets in, the pellets turn into fluffy "sawdust" as they absorb moisture. When I notice they can't absorb any more moisture, I replace the pellets. For the chickens, I've been stirring the pellets up after scooping the biggest piles of poop off the top and then stirring it up. And the chickens will also stir up the bedding on their own.
Before now, the last time we had chickens was when I was a kid - back then the coop floor was dirt as well as the run. Nothing fancy. I recently saw the same set up of dirt flooring at someone else's place and it wasn't a problem.
Have you looked at the deep litter method, where you let the chickens scratch up the poop and then you only have to clean it up a few times a year? Here's some links about that: http://www.backyardchickens.com/a/deep-litter-method-the-easiest-way-to-deal-with-chicken-litter-dlm http://www.purelypoultry.com/blog/the-deep-litter-method/
Using food grade diatomaceous earth in the bedding can not only help keep it drier but also helps with external parasite control for your chickens.
I just read some posts from someone that uses a zeolite product called Sweet PDZ in their coop and they have a lot of chickens in there. Said she built a tray that sits under the roosts - since they always poop all night when they are roosting - and she fills the tray with the Sweet PDZ. The poop falls in and she scoops it out, said there is pretty minimal effort and smell. I am looking at trying it since she said it worked so well for her. Here is the link for that product, perhaps they have something similar where you are at. http://www.sweetpdz.com/our-products.html
You don't have to put anything down in the run unless you want to. Like if it gets too muddy with rain/snow. Then you could put down sand or bedding material.