Shortcircuit first I gave a really short answer to your last question so here goes the long answer.
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The code section requiring the ground rod for a separate structure is this one.
part of 250.32
250.32 Two or More Buildings or Structures Supplied from a Common Service.
(A) Grounding Electrode. Where two or more buildings or structures are supplied from a common ac service by a feeder(s) or branch circuit(s), the grounding electrode(s) required in Part III of this article at each building or structure shall be connected in the manner specified in 250.32(B) or (C). Where there are no existing grounding electrodes, the grounding electrode(s) required in Part III of this article shall be installed.
Exception: A grounding electrode at separate buildings or structures shall not be required where only one branch circuit supplies the building or structure and the branch circuit includes an equipment grounding conductor for grounding the conductive non–current-carrying parts of all equipment.
The circuit feeding the outlet the trailer plugs in is
one branch circuit, so the exception is available to you if you want.
Yes you can add a ground rod any place in an electrical system you want and connect it to the equipment grounding conductor at a place of your choosing.
250.54 Supplementary Grounding Electrodes.
Supplementary grounding electrodes shall be permitted to be connected to the equipment grounding conductors specified in 250.118 and shall not be required to comply with the electrode bonding requirements of 250.50 or 250.53(C) or the resistance requirements of 250.56, but the earth shall not be used as the sole equipment grounding conductor.
Will it really do anything for lightning protection?
That is beyond my knowledge but it certainly will not hurt anything.
