It has been quite a while since I last did an update. Things are going well overall! We ended up with a total of 38 chickens by the end of the Phoenix summer. We lost 2 of the older girls to the heat, but all of the new pullets made it through the summer with apparent ease. We did lose one chick to a water bucket that broke and fell on her. 
The current issue is with RATS!!! We just watch them on the chicken run cameras every night. They have not harmed any chickens at this point, but they are eating a TON of food and have been stealing eggs and hiding them (though not well). We have been attempting to trap them, but with no success. We also ordered a couple of the large Grandpa's chicken feeders and will be replacing the feed buckets with those. We will also be shoveling out any and all feed from the ground of the run this weekend.
Unfortunately, because the chickens have the entire side yard as their run, there is NO way we can rat proof the run, so eradication is the best bet. Hopefully, eliminating access to the feed will move things in the right direction. Any other ideas or pointers to help with the rat problem are greatly appreciated.