We recently had a diversion because the airport was giving cat III but the A/C was only cat II. The RVR's were above cat II minima but the Cap't did not think he was allowed to make a approach!
Well, because the Captain was wrong.
If a higher grade approach is available (e.g airport is CatIIIb capable) it should be available to operators capable of using it. It will safe-guard against sudden WX changes. But if the RVR is sufficient for the best approach YOU are capable of exisits (CatII in this cas) there is nothing stopping you commencing the approach to your minima.
As to the opening question, the opposite is also true. If only CatII is available, only to CatII do you go!
I think the airport was at fault in this case, if everything for CatIII was available, they should have said so.