For guided ice climbing, you might want to check Sanirang for possibilities. http://sanirang.net/guiding/
They're based near Seoul but maybe you could work out a custom deal.
As for spelunking..... there are a few caves around the east coast... but I've never heard of any real exploring opportunities through them. We're talking neon-lit walkways. Still fairly interesting..... but not quite as exciting as you might be after. Again, ask Peter at Sanirang, he might know. Perhaps another community member will pipe in with a better idea than I have.