You probably want one big enuff to hold 2-3 lbs of specialty grains. I dont have the actual simensions of the size of the one I use but if I had to guess it's probably 8 inches wide by maybe 12 inches long.
I'm pretty sure that "coarse" refers to the size of the holes in the mesh. The more coarse the bigger the holes are in the mesh.
I havent thrown mine in a dishwasher, I'm pretty paranoid about using any detergents on any of my brewing stuff that might even have the slightest chance of absorbing them. When I first bought it it was all nice and white, but with in VERY short order turned a brownish color with a tinge of green to it. Thats from the grains and hops so it's nothing to be worried about, but when I want to wash it out a bit, I rinse it under the faucet in warm water to get all the major chunks off it. Then throw it in a pot of boiling water and let it boil/soak for a while. It's doesnt look like it reallys helps much until you pull it out and go to dump out the water and notice that the water is now stained brown. The guy at the LHBS said it's normal and wasnt anything to worry about and then said if it really bothered me I could always come in and get a new one