Hey MM, yeah I actually live in Korea now, on the coast in Gangwondo.
1) Although Korea is definitely popular with Koreans, and increasingly Japanese and Chinese tourists, it's still pretty easy to get away from the crowds even on the weekends. Personally, I like the Jeju Olle Trails that pretty much circle the island. I've only hiked a couple of them, but you get a wide variety of nice scenery. They can get popular with tour groups, but I think it's easy enough to get away from them.
2) I really like Jeju, although I don't watch Korean movies so I can't say how it compares. Koreans often compare it to Hawaii, which is ridiculous. I've been to Jeju twice and Hawaii 3 times and they're nothing alike. Jeju is still beautiful, though, just a different kind of beauty.
3) I'd recommend Udo, which is a small island off the east coast of Jeju. You can catch a 15 minute ferry from Seongsan, which also has a giant seaside caldera worth climbing up. When you arrive on Udo you can rent bicycles or scooters as soon as you get off the ferry (for the scooters you might need a Korean or international license - I'm not sure as I just got a bicycle).
I'd also suggest staying in Seogwipo rather than Jeju City as it's more laid back and I prefer the scenery on the south coast. You can actually catch a bus to Seogwipo right outside the arrival hall at the airport that takes a little under an hour I think and costs maybe 5,000 won (I can't remember exactly).
There are actually 2 main waterfalls in Seogwipo. I can't remember their names, but there's one just south of town by the harbour, and the other is a little east of the harbour. You can walk to both of them. They're both nice. but I doubt they'll blow you away.
I went to Pyoseon Beach in January and it looked like a nice beach, but obviously it was too cold to go swimming. I hiked the Jeju Olle Trail from there west towards Namwon. On that section you go up to the top of one of the oreums. Worth doing.
I know Manjanggul (a really long cave in the east) is quite popular, but it didn't really do anything for me. It's just a long underground tunnel really. I'd spend my limited time elsewhere if I were you.
If you have any other questions, let me know,
Jason