1. lots of consulting firms in Korea, all the major ones, but a base requirement is that you are billingual.
2. you want to come to Korea as an expat, to be hired in Korea under that respective subsiderary, may give you less benefits and salary, that isn't always the case, but its happens.
3. Your experience seems fine; in Korea, the interview and case study holds a lot of weight, but don't expect that since you're a foreigner you'll get any special consideration because everyone at the top firms are all billigual at a professional level.
4. I'd suggest using a headhunter and see where that leads you, but of course applying directly is good too because nearly all the firms have open times for applications.
anyway(s), good luck!