You are doing too much.
Jeju is a great change of pace from Seoul, BUT you must have an international driver's license to rent a car and drive OR pay 100-150 USD for a private taxi tour (maximum four passengers). If you do the taxi tour, it is economical because the island is so big and they will take you everywhere. You can arrange your flights to arrive at the airport at 9 and they pick you up there. They will drive you until 7 and drop you off at hotel. Then, the next day, again 9-7 and drop you off at airport for your 8 PM flight back to Gimpo. Public transportation in Jeju is not reliable and would take HOURS.
Busan is a cool city, but honestly, you won't feel much difference from Seoul in two days. You should spend more time in Seoul.
Also, for skiing, it is not worth an overnight stay. The options are expensive and there isn't much to do there at night. Instead, the ski resorts offer shuttle buses from Seoul and you can do a full day trip. Save your time. As far as which resort is good.....they are all crowded, busy, similarly priced and similar levels of difficulty. It's not worth a two day trip.
For New Year's Eve, the only "big event" is in Jongno near the Bell Tower. Most Koreans just go to bars or don't even celebrate. If they do celebrate, they always go to the east coast to watch the first sunrise.
But, all in all, I think you are trying to do too much in a short time. I understand you want to see the country, but you can definitely spend 3-4 days in Seoul and still have a lot to do.
I am happy to help you out and provide advice, but Seoul is so big and full of things to do that I can't possibly tell you everything. Let me know what you and your group like to do, ages, etc.