You can't cut the cross piece out from under the roof. That's what's keeping it from twisting and it provides additional support and stability.
For starters, you need a level and 3 12' 2x6. Measure level under the supports from the inside to the inside of the post. Cut one board. Toenail it in place right under the supports. Measure the back side the same way. Nail it in. On the back, (the front is the higher roof side) where the run will be, measure at ground level the same way. Paint the board where it makes contact with the ground and nail it in place. You are going to want the nesting boxes outside on either the front or back to save inside room. If it's on the front, than you don't have to go inside to gather eggs. If it's on the back, it will give them a little protected area outside to sit under. Find some detailed instructions under this section on how to build them. You will want 2-3 nesting boxes max. You can go with something simple inside but it will cut into your floor space and the # of hens it can house. It's the right size for about 8 hens right now. I'm assuming you are in GA so the open lattice sides can work well for you. I would add wire on the inside just to beef it up a bit.
Start there. You will end up adding a door, a pop door, maybe a window but it will be a good first building project.
