First of all, I recommend bringing at least $1000 USD of spending cash for the first month. Pack toiletries as if you were going on vacation, just so you have enough until you are fully settled into your apartment.
Between Costco, Gmarket, iHerb, and foreign marts, I’ve found that most things can be purchased here -especially if you’re willing to pay more. I believe in the old adage that if Koreans can live without it their whole lives, you can probably live without it for a year, but I’ll still humor your question with an educated response.
Bedding: I have never wasted precious luggage space this way, but friends swear by packing their favorite fleece blankets, jersey sheets, or fitted/flat sheets. All these items can be extremely difficult to find in Korea.
Birth control: If you prefer a certain brand of condoms or pills, stock up in your home country and bring them over.