Re: Bottom Board use & management
Welcome to Beesource!
You will find that beekeepers disagree on the value of screened bottom boards (SBB)! So there are no absolute answers.
If you are going to use SBBs, then #8 hardware cloth is the correct size. If you goal with the SBB is ventilation management, then you may wish to close it partially/fully off in colder weather. If you are using it for pest control, perhaps with an oil tray, then the tray already acts like a partial block of the screen from a ventilation perspective.
Really, before you can get good answers to most of your questions, you need to decide why you are using the SBBs in the first place.
Personally, I have built SBBs into my bottom boards, but with space for a tray below the screen, and then a solid bottom below the tray. I plan to leave the closure for the tray area permanently in place, except to periodically access the tray.
--- Practical reality trumps philosophy!