I live in Japan and have never hiked in Korea, but Korea's climate and terrain are almost identical to Japan's.
Not only will it be hot, but it will be incredibly humid. You'll want at least a light, loose, quick drying t-shirt. June however is very temperamental, especially in the last ten years, so it could also be very cold and higher elevations (above 2,000 meters) could still have a lot of snow. Korea is colder than the same latitude in Japan, so prepare for possible cold temperatures.
Water-breathable rainwear only goes so far in hot humid conditions. Most likely you'll overheat as you're climbing. Bring the rainwear for rest stops and cold ridges and such, but very often it is much more comfortable and efficient just to let yourself get wet, as long as you don't get cold. Your body heat will dry off your clothes quickly.
If the monsoon does hit, you can get days and days of unending rain, though usually just drizzles and not downpours. Mud will be a very big factor in your walks. Some areas will probably have leeches (nothing to worry about except when you're camping). I'd try to wear light shoes with good traction. In the monsoon your feet will get wet no matter what you wear. I almost always just wear light runners with mesh tops that dry out quickly, although with mud mesh may not be the best.
The mountains in Japan and Korea are STEEP! Try to be in good shape when you get there.
I don't know about Korea, but in Japan the authorities tend to be pretty lax about camping, even when there are rules against it. As long as you're tidy and considerate I've never had a problem with camping where I wanted, as long as it wasn't on protected wildlife land.
If the hut system there is the same as Japan's probably you don't need to worry about food much. You can get whatever basics you need at the huts, and great, healthy prepared meals. The huts also make making friends very easy. You'll probably see Koreans at their best and most charming there. Koreans absolutely love hiking and are a very friendly people, so the huts would be a great place to meet them.
I do have to say though... as a Backpacking Light member your list sounds a bit heavy and conventional for the folks around here. I'd look around at some of the articles on packs, sleeping methods, shelter, and cooking that BPL offers, to get an idea of how to lighten up the packs and become more efficient with each item you use.