I went to Korea during early winter. It wasn't snowing yet that time but it was freezing cold. If you want to buy kpop albums/merchs, go to Dongdaemun Market instead of Namdaemun Market. The albums are a lot cheaper there. Since it's winter, go skiing at Yongin, it would be so much fun there! Namsam tower is a must. I bet it would look even pretty at night but I went there on morning but it's still a beautiful sight to see :)
Didn't get the chance to go to Lotte world but I went to Neverland and I must say that, if you are planning to go there, then don't add any activity that there because there are A LOT of things for you to visit in Neverland.
If you go to visit Nami island, try out the hotteok there. It's a pancake and boy, it's soooooooooo delicious :)

I'm going to Korea next week and I've planned a few things already. (I'll write some of those places I want to go but can't)
Firstly, You should go to the shopping places (Like dongdaemun and others)
Secondly, for me, I'm going to the palaces and Hanok Village (Just love Joseon dramas)
Thirdly, Namsan tower and Hankang River
Fourthly, Petite france (Secret garden and runningman set!)
Fifth, Nami island!
If you have more time, go to Jeju.
Tip: From what I read, Lotte World is quite expensive, because not only you need to pay for the entrance fee but also the rides. So its quite costly.
If you are legal, go to clubs in Hongdae, a lot of celebs love to go there. (If your illegal, its fine. Korean clubs aren't strict, they won't check your ID)