1. The original meaning of the is debated - to say the least. BTN lists a few interpretations, but I found two more, about one of which I'd like to know more. (The other one is just funny: katha = complete + ruina = fall; this from Legenda Aurea by Jacobus de Voragine.)
Now the other one says, it means "ever pure" and refers to the older form Aikaterine. Now I wonder: what is the "ever-" part supposed to be? "ai"? Or "ei"? My Greek dictionary doesn't give a hint.
2. I also found that Aikaterine is still the more popular form of the name in Greece today. Can you confirm this?