Could you update your info to include a location? I'm not sure what ABQ means but am assuming NM is New Mexico. AZ ashe - Arizona Ash? Yes, a messy tree.
Most trees will root deep given proper care although some just want to play on the surface no matter what. Lawn sprinklers will not provide proper watering for trees; grass only needs to have moisture to the bottom of the roots - less than 6 inches down. If you want your tree to root 6 inches deep, putting it in a lawn will accomplish that. Trees need deep watering every week to 2 weeks, depending upon weather. That means getting a hose out or having a double irrigation system - one for trees and one for lawn. The roots will follow the moisture down. If there is a break in the moisture, the roots will not grow through it. That means consistency - if you are watering every two weeks, put it on your calendar. The worst you can do to a tree in the desert is leave a dry spot in the root zone.
The alternative is to grow trees that are made for the desert climate: Russian Olives, Mesquite, Palo Verde, Locust....
The problem with evergreens is that they grow slooooowly. Deciduous trees are faster. If you want shade soon, you may have to plant another deciduous tree and put up with the mess of leaves and possible fruit.