Is Late December Early Jan a good time period to travel to Korea?

Hi all, this is my first time posting here. I'm planning a trip to Korea around late December to Early of Jan to celebrate Christmas and New Years there. I've heard from alot of people that it'll be very cold, but I plan to go skiing so it should be fine for me. My travel period will be around 23 Dec to 4 Jan.

Anything I need to watch out for? And how cold does it actually get during that period? And is there anything noteworthy (For eg, during Spring, cherry blossoms are noteworthy)? And do Koreans celebrate Christmas and New Years? In my country during New Years, the sky will always be light up with full of fireworks for a few hours, and i was hoping to see it in Korea. And how much money do you guys think I should prepare, excluding accommodations.

Also, should i stay longer in Busan or Seoul? I plan to go to a skiing resort for maybe 3d2n and the rest will be spend on both Busan and Korea, haven't really planned out my itinerary yet as I'm still unsure where to stay longer. Sorry, I know this posts is very messy, I just had a few questions and decided to might as well clump it all together.