I can't predict when the leaves will change colour this fall, but if you are planning to visit one NP, your best bet is probably to do it as late as possible (i.e. near the end of your trip) and to go to one of the northern parks. You'll probably be here too early for the central and southern parks to be changing colour, but you might get lucky with the northern ones.
My guess is that Seoraksan or Odaesan (just south of Seoraksan, outside Gangneung) would be your best bets. They're both very nice and worth checking out, but Seoraksan is way more popular, and the trails can feel like you're lining up at the bank. Whatever park you choose, I'd highly recommend going during the week, rather than on the weekend.
As for your 2nd question, I'm not sure if I understand your concern. Are you worried about hiking alone, or do you just want people to talk with? If you're worried, I'd say that there's not much to worry about. The national parks in Korea are very well done and the trails are usually in excellent condition, and most of the trails also have some English signage. The only place you have to worry about bears in Korea would be Jirisan in the south, but even then I think it's very rare for people to see them.
I've hiked dozens of times all over Korea, most of the time by myself, and never had a problem. Koreans love hiking, so unless you go to a really small out-of-the-way non-national park, you'll encounter many other hikers.
Jason
