Here are pics of my setup. Id say its one instance when the cheap prefab coop/houses are perfect. If you notice, mine has trays on each section. But the height is just fine for bantams, but would be too low for a full size. But if you got one w just a bit more height, you could easily build out a bottom edge to hold in a couple trays (since the coop portion usually already has a tray on all the designs).
I also bought a very thick vinyl shower curtain and cut it to create a skirt on the interior of the "run" so they wouldn't kick out their food or bedding. That has made a world of difference, house stays very clean.
As for bedding, i tried so many things and they were an utter failure. Finally, i used horse pellets that are watered down (til water covers dry pellets), let them expand, and it turns into a very nice smelling, soft, not dusty, absorbent material.
Food and water cannot be going everywhere like outside, so i splurged and am so happy w my Royal Rooster waterer and feeder. No mess, clean for the birds, and no waste either.
I let them out to play and sunbathe in the morning and the late afternoon. They do like their little house though, so they only stay out about 20 mins and then put themselves back in. If its just that long, i dont worry about diapers. If its longer, then i did get diapers for 2, need to order 2 more (the chicks are still growing, so the diapers are just slightly big for now).