Your two questions are easy to answer. Valves should be left open so that any residual water doesn't damage it. The PRV is for your household water and has nothing to do with you irrigation system.
Now to the rest of the problem. Unfortunately, many lawn systems are installed by homeowners who know nothing about what they are doing. Yours may or may not be one of those. Among the problems in trying to diagnose you problem are we can tell what you have, and you don't know enough about what you should have to tell us. My best advise is to call in a lawn sprinkler system installation company, have them examine your system, show what you have, advise you if there are things that you don't have but should have, install or repair parts as needed, and show you how to operate your system.