What exactly is different with the different ILS categories except for the decision height? Speeds, etc. all remain the same?
CAT III approaches are generally flown at half the stall speed
Speed depends on the aircraft, its configuration and the weather.
Is it ever possible that your RVR corresponds to a certain category, but your DH is too low for that ILS category?
Imagine a thin layer of fog at 150ft AGL. DH is insufficient for CAT I but RVR will be plenty once you pass through the fog.
Everything > CAT I is reserved for the professional world. Flying at CAT I minima in a SEP can be quite scary. Synthetic vision and radar altimeters are good things to have.