Put it where the ground is high so water doesn't collect in it. It is best to put it in the shade, and where it will get a good breeze in summer, as chickens have a lot of trouble dealing with summer heat. You can have a solid wass facing where your winter winds come from, but you might find you need to put a window (can be just an opening covered in wire mesh) there in the summer for the breeze.
If the run is predator proof, you don't need a coop door for the chickens, unless it's to block winter air from blowing on the roost . I've never heard of using a doggy door; I would think it would pull their feathers out, but maybe not. I've seen those clear vinyl strips used, though.
Chickens often go out in the rain; you can let them choose. Mine don't go in unless it's really flooding out. Most chickens don't like to walk on snow, though, so an outdoor run that's at least partially covered will encourage them to go outside. If you get much snow, you'll probably find tarps won't work well.
Build it as big as you can. There are plenty of uses for extra coop space -- keeps them peaceful if they want to be in because of weather, storage, space to have feed and water inside so there is no worry about feed getting wet, maybe a place to add a sick pen or broody pen or brooder, etc. I have an oversized coop partly because I used to have a lot more chickens, but the extra space makes everything SO much more convenient, and gives them room to get away from each other if they are inside and awake. It also gives them the sideways room they need to fly off the roost safely, without hitting a wall.
