If an opponent plays a card such as Local Unrest and targets my last ground force on the planet, that planet is turned neutral. When that happens, do I lose any PDS's and Space Docks on that planet, or do they remain? If my forces remain there, do I regain control when I land ground forces there again?
The Space Docks and PDS units are destroyed.
From page 3 of the official FAQ version 2.3
Q: If a planet contains one Ground Force, one Space Dock, and one PDS, and “Local Unrest” is played on that planet, are the Space Dock and PDS destroyed?
A: Yes. The Space Dock and PDS would be destroyed due to the planet changing ownership (becoming neutral).
And for good measure, on page 12 of the rulebook:
If a PDS unit lands alone on a neutral or enemy planet, it is immediately destroyed.