1. Did you receive a dongle with it? If not, you may have gotten a pirated copy which may explain problems, although I don't think the problem you described lies with Logic. If you don't have a dongle, then I would suggest just purchasing a new software. Already, I think the seller cheated you if it's an original software. The manual is one thick 3-ringed folder with nice padded covers. It is quite well written. That should have been included in the package he sold you - unless it is pirated. Also, is it Logic Audio Platinum, Gold or Silver?
2. The latest PC version is 5.5.1. It's no longer supported and I'm not sure if you can still find the update somewhere. You can try to google for it.
3. Even then, the problem doesn't not seem to be Logic. Since you can trigger the GM sounds, it is working perfectly. To trigger the other sound banks in Korg, you have to:
(a) download the template of Korg. I've tried very hard to look for you by googling, but I can't find it. If you have more time, you should search.
(b) even if you don't have the template, you can still make your own! As I said previously, double click on one midi track; this will access the instrument environment; on the top bar, click New -> Instruments; you get a new icon; on the left side of the environment, you can select the midi bank (midi numbers etc) that correspond with your Korg. You can access ALL the sound banks in your Korg! Shouldn't be a problem.
As for the gain level, it doesn't sound like a Logic problem either. What is the hardware you are using (soundcard)? How did you connect your Korg to your PC?
I feel the problem doesn't lie on Logic. Otherwise, you'll be purchasing another software and the same problem will exist (unless you got cheated and purchased a pirated copy). If you want me to guide you step by step on how to set up the instrument environment, pm me. Make sure you have your Korg manual around to access the midi numbers of the patches.