What are my options for installing a water softener on the opposite side of the house from the main water supply?
If it were me, I would definitely put the softener in the garage.  Protecting it from weather, critters, and extreme temperature variation seem reason enough.  But to also have it where you can easily monitor it daily seems reason enough again to locate it in the garage.

But the water meter looks like it is above ground:  Maybe the climate is safe against freezing?  I am under the impression that all of Arizona is desert climate, which means below freezing temperatures are possible overnight most days of the year.

You only need to run one pipe from the water meter to the softener.  From inside the garage, you should be able to connect into the system more directly than going back to the meter.  In fact, would the water heater be in there too?  (I don't see an exhaust for one maybe because there is not enough resolution in the photos.)  If it is, you can put a T fitting onto the cold feed into the water heater and distribute softened water from there.  Cap off the end near the water meter.

As for digging and burying in the front yard or running it under the house, I would expect running under the house is easier and cheaper.  Does it have a crawlspace?