Odds of me getting leave for Christmas in Korea?

Not too good. Literally everyone in your unit will try to get leave on Christmas. Your unit needs to maintain a certain percentage of it's strength year round. You might have a special case considering the deaths in your family, but that's assuming that they died relatively recently and you're planning on attending their memorials. My grandfather actually died a few days after I reported to Korea and I wasn't able to get leave approved to go to his memorial. The logic behind this was that my unit was under strength on NCOs, we were about to go to the field and he wasn't immediate family. So, I had to go to the field within 3 days of arriving at my unit, all while my grandfather's funeral service was going on.

Chances are pretty good. Mission dependent of course. Have you been contacted with your sponsor? If so, you should inform them of your intent immediately. If not, as soon as you get to your gaining, notify your team Sergeant/squad leader of your intent so that they can look at the training calendar and forecast the holiday block leave dates. Nonetheless the sooner you start inquiring to your leadership of your intent the more likely the chances of it becoming a reality. Good luck and my condolences to you bro.

Your situation is unique SPC (Join to see); inform your sponsor at once and; take 10 days only. As Top has confirmed, you will have to pay your way both to and fro; a return ticket is cheaper. I am certain, depending on mission, your outfit will approve your leave. Enjoy it.