Living in the north as you do I would say to use the screened bottom with the sticky paper. I use them here in SC to help cool the bees during the warm season. I do not even use the sticky paper. It seems to help to cool the colonies and also rid the colony of a few of the mites.

If the hives are off the ground, there is no need to ever close up the screened bottom board except to do a mite count a few times a year. You can leave the screened bottom board open all winter. The bees have a harder time in winter with moisture then they seem to have with cold. I never use a solid bottom board, the hives are always fully open through the screened bottom board (the bottoms of my hives are about 1.5 feet above the ground on stands), and we get occasional spells of -10 degree F winter temps. I do use straw bales as windbreaks on the north and west sides of the hives, but the screened bottoms are always open.

I took off most of my solid bottom boards this spring. I was running screened bottom board on top of solid bottom boards but the space in beteween them just seemed to be a place to catch debris and encourage wax moths.I'm not sure if I'll winter with them yet. I do provide plenty of ventilation so moisture isn't a problem in the winter. I live in the north western corner of MA