You can mix chicks of different ages as long as all are less than a month old. They are unlikely to have lice/mites or other problems if they are all from the same feed store.
Your planned coop and run are really on the small side. Your big enemy is probably going to be heat. An enclosed coop may not be the way to go if you are in a desert area, particularly a small one. The important things are going to be breeze and shade -- and even being able to wet the ground because of heat.
Here are some ideas for living spaces for chickens in hot climates:
https://www.backyardchickens.com/t/163417/please-show-me-your-hot-weather-coops
Your planned coop and run are really on the small side. Your big enemy is probably going to be heat. An enclosed coop may not be the way to go if you are in a desert area, particularly a small one. The important things are going to be breeze and shade -- and even being able to wet the ground because of heat.
Here are some ideas for living spaces for chickens in hot climates:
https://www.backyardchickens.com/t/163417/please-show-me-your-hot-weather-coops
from Ohio. So glad to have you aboard. 
