You might consider a 8x16 instead of a 12' square. A couple of reasons. Most building materials come in 8' and 4' sections. 12' lengths are not bad, but 8' increments could mean less cutting and waste. Just something to consider.
The other reason. The wider the building the heavier your rafters and general roofing needs to be to support that wider span. With a 8' width as opposed to 12', you can get by with less expensive wood or, with the same wood, have a much stronger roof.
Slope your roof away from the run. You do not want the rainwater runoff going into your run. It will be wet and muddy enough.
I suggest as much ventilation as you can stand. It gets hot in Kentucky and chickens do not do hot real well, since they are wearing a down coat. You need to get rid of the summer heat. As others have said, cold is not your enemy, humidity and ammonia is. You need good ventilation in the winter to get rid of both. You do not want breezes on your birds on the roosts in the winter. Wind chill is bad. Your ventilation needs ot be well above the roosts so any breezr passes over their heads.
Have overhangs on your roof and put ventilation openings under the overhangs. Here are a couple of shots that kind of show what I did. I built mine on the end of a shed. You can see that the one wall is on the inside, so you can put plenty of ventilation there without rainwater coming in. You might not be able to tell, but hardware cloth covers all those openings.
