Re: Screened Bottoms
There are several different designs. One of those is just an open screen. The concept is that mites and beetles will fall to the ground. Those fallen mites will probably die, but adult SHB can fly, and immature SHB pupate in soil.
Another design has an "oil tray" underneath the screen to kill anything that gets into the oil. Of course, you need to arrange things so the bees cannot get into the oil.
#7 or #8 hardware cloth is the right size to use. Some bees can get through 1/4" hardware cloth. See post #8 of this thread for more info:
You can find free plans for a screened bottom board with an optional "closure" board at the bottom here:
The plans do not show enough vertical space for an oil tray, but you can simply increase the distance between the screen and the lower closure board to make space for a tray. You will then need to buy or build a tray to hold the oil, and make a wood "plug" to block bee access to the tray while still allowing for maintenance.
The end result is very similar to Country Rubes pre-made version:
One option for a pre-made metal tray is a half sheet cake pan (13"x18") similar to this:
Plastic cafeteria/lunchroom trays are also available in a 12"x18" size, generally at a lower price than cake pans.
USDA Zone 7A Elevation 1400 ft