Not just the numbers, but do I have the idea right?
Basically, yes.
What exactly is different with the different ILS categories except for the decision height? Speeds, etc. all remain the same?
The training, equipment, sometimes wind (due A/P limitations), DH, airport equipment and status, runways slope, RVR just to name a few.
Do you find the DH if you're doing CATII, or CATIII on the approach plate itself?
For CAT II, these are published on the plates we use but CAT III minima are operator specific. These are supplied by your company and come as a result of specific Low Visibility Operations training.
Is it ever possible that your RVR corresponds to a certain category, but your DH is too low for that ILS category?
I think you have this arse about face. Assuming you have a "standard" CAT III runway and an appropriately equipped and trained crew, the RVR defines the lowest DH that can be used and by virtue of that, the type of approaches that can be flown. So if the RVR is greater than 550m, a crew could fly a CAT I, II, IIIA or IIIB approach. Drop the RVR by 100m to 500m and a CAT I approach is (generally) no longer possible. Drop the RVR to less than 300m and a CAT III approach has to be flown.
Does that make sense?
PM