If I was doing this, and someday maybe I will as I have 600' to the road from my nearest power. I would run a #12 UF in a 3/4" PVC conduit from a 15A breaker. This is assuming the Mighty Mule opener uses a battery and the 120v is just to run the charger. I would not worry about voltage drop for reasons stated later.
A 500' run of #12 is about 2 ohms so the voltage drop would be 2 volts for each amp flowing in the circuit. A short circuit at the far end would draw 60 amps so the 15A breaker will protect the circuit from faults. You could operate some lights with this circuit but would not be able to run any electric motors of any size. A small "wall wart" type battery charger only draws milliamps, thus does not have a voltage drop problem.
The code specifies that a voltage drop not exceeding 5% provides a "reasonable efficiency of operation". I was taught that this was a good practice but NOT required by code as long as a fault would trip the breaker. I.E. low voltage is not a fire or personnel safety issue. However this is not a universal opinion. The final answer would rest with the Inspector, I.E. "The Authority Having Jurisdiction". Some inspectors will consider this to be a requirement, some not.
I would, as Eddie stated, have to slide the conduit on and only be using it as a mechanical protection from the large number of rocks here. You would never be able to pull that UF through 500' of conduit. Around here I would not use romex in the PVC conduit as sooner or later the conduit will fill with water (long story on how this happens). The UF is rated for wet conditions. I would bury it 24" deep as I will need a trencher for this so I might as well run it that depth.
As far as running decorative lighting, voltage drop is not a real concern, the lights will just be dimmer and last longer.
