Either controller is good. Check number of zones by opening up the old controller and seeing how many ports are used. If you buy a modular controller, make sure you buy enough modules.
Do it yourself is simple. Unplug it, remove 1 wire at a time and label it (common, main, zone 1, 2, etc.), get new controller and mount it (make sure the wires will reach to their intended port before mounting), wire in the electric and plug in and program.
Back flow devices protect the public water supply from potential hazards. Most states, including TX require them. They are important, they can malfunction or freze, and most states require they be checked once a year.
MO might require an irrigator to be licensed, TX does. If he can produce one that should be enough, if you aren't satisfied with him you can take him to the licensing state agency and hold his feet to the fire.