First to answer all of your questions: Yes they have Costco in Korea and you can use your Costco card from North America in Korea.(don't forget them at home when you come visit)
The first week that I got to Korea the principle at my school mentioned that we should go to Costco to get as he called it, "your people food." We talked on and off about it and finally we decided at lunch that we should go after work that day. We turned out to be the principle, the co-teacher, the 5 year old teacher and her 3 year old son and I.
Let me tell you if you are looking for adventure just mention a Costco trip and adventure is right there with it. The first part was travelling with a three year old, thankfully one of the others lend him their phone to keep him occupied, unfortunately he occupied himself with a game that I called Copy Cat(I don't know the actual name). Calling it a game is some what misleading in that it is not a game at all, it is a cat that stands there and repeats, in a different tone, any sound that it hears. This caused much yelling and squealing that I got to hear not once but twice! This in it's self was not terrible but it was compounded by taking a wrong turn and getting stuck in rush hour traffic making a 45 minute trip take 2 hours. At least one of us had a good trip.
After a quick stop by the food court(very similar to home) to grab some delicious and cheap pizza and a bulgogi bake(think lots of beef and lots of cheese wrapped in dough Costco size) we were shopping.
I had a number of items on my list but it was filled for the most part with suggestions of what I wanted.
This is a picture of what I ended up coming home with:
As you can see I bought only the essentials, a 10 pound bag of pancake mix(10000), 5 pound of margarine(10000), cheese(great deal)(11000 each), a dozen muffins(8000), 1.3 kg of oatmeal(13000) alfredo sauce(10000), tomato sauce(11000), jam(11000), peanut butter(expensive but essential)(15000) and a few other things to balance out the rest.(numbers = price in won)
Altogether I ended up spending 140,000 won, completely blowing any budget I had but the hope is that the majority of this food will last me until my next Costco trip in 6 months.