I used linoleum on top of the plywood also. We didn't think ahead enough to frame on top of the linoleum. Instead we used some free 2x6's we had to line the border of the coop and cover the edges of the linoleum. Then we painted it in too.
You can see a picture in the link in my signature.
We got the linoleum remnant half price, and it makes it so easy to clean - and easy to stir the shavings - whatever needs to be done.
In regard to mildew, etc., i washed the walls, etc. down with bleach and then painted them before we put chickens inside.