Just reread all the posts...hard to read long posts on my phone, but you may be able to get away with not installing a ground rod(s) if the RV box is not attached to the building.
Honestly, if the cord to your RV is long enough that you would be able to mount the 30 A 120V outlet on the shed, I would do that. The cost of a ground rod and a little bit of #6 would probably be less than if you ran the feed to a free standing pedestal then back underground to feed the shed.
Remember...Romex may not be allowed in that shed depending what code cycle you are under.
Even if you are on the 2011 code cycle, you still may not be able to run romex without protection from physical damage. Also, GFI protection for any outlets in the shed would be required.
and as a little side note: Where is the GFI breaker for that duplex recept in this panel?