Places to play - Papa Board

PAPA Board is located in Changwon in the South of South Korea.
It looks like they take the cafe side of the board game cafe part very seriously and it seems a nice place to play.
As you can see they have a LOT of plastic hammers and are clearly worth a visit if you are in the area.

I have never been there though and can't offer much advice on how to even find the place. According to their directions they are very close to the South East corner big intersection/rotary in the middle of this map. 
If you are near Changwon I would seek out t a little more information before trying to find this place. Hopefully they show up on Wizards store locator soon. 

Korean Singles

Listening to the latest episode of the EH Team Jeremey Schofield lamented how hard it was to find Korean singles of the magical card variety. The other varieties are rather easier to find.

The Korean Ashiok was printed to incorrectly give plus 1 instead of plus 2 for the first ability. This has caused there to be extra demand for the Korean version even when it's not in foil. I remember the Portuguese Meloku made 2/2 tokens instead of 1/1's and to me that sounds like a much cooler misprint. 

To buy anything online in Korea you usually need a Korean ID number while only using Internet Explorer and downloading a bunch of weird extensions. I don't think you will get much joy from using the shopping cart tools on their sites and would suggest trying an email instead. Most of the store owners I have met speak fairly good English but may be a little intimidated on the phone and I wouldn't trust them to get all the details right regarding address and payment. 
Koreans tend to be much better at reading and writing English than speaking and listening, and an email gives them something they can examine multiple times.

There are 5 major online Korean shops that I know of:

http://cardcastle.co.kr/

http://www.divedice.com/
I have ordered from them before and got great service but that was domestic shipping using a Korean ID.

http://www.mtgkorea.com/
This is the online shop for Dalmuti's in Kyodae. Very good singles selection but I can't see any option to ship overseas. Does not require a Korean ID to register which is a big plus.

http://abouttcg.com/

http://k-mtg.com/
K-MTG formerly known as Playfirst sell singles worldwide. This is the most foreign friendly website out there and a lot of people recommend them. I am a little hesitant as the only time I ordered from them they stuffed up my address thanks to lazy handwriting. I was lucky to get it thanks to a diligent courier.

http://redmtg.com/xe/
This is the online store for Red Dice in Seoul.

Korean singles are not cheap and Korean players know the worth of their cards. They are not oblivious to the fact certain Korean cards (foils of legacy/modern staples) will attract a premium. Japan also has quite expensive prices for singles but is also well known for being a great place to sell cards. Korea,  not so much since most of the demand is for standard cards (Innistrad represents the first rotation out of modern Korean cards). Standard cards that aren't the top tier are hard to move and that is not just in Korea. Look at the buylists for Starcity and you will see the prices they offer is quite smaller for type 2 stuff than what you would expect.
It's possible that there might be decent prices available for those cards that have rotated out of standard because there isn't a lot of Modern and virtually no Legacy. This is something I will try and check out because I am only interested in Modern as a constructed format but I found it incredibly hard to get Liliana's of the Veil just a few weeks ago and ended up getting them from America.

I have been contacted a few times from people online asking me to get them certain Korean cards or sealed product. I'm sorry but I am not interested in doing that despite promises of payment.
I have various reasons for saying no, the main one being that I am very selfish. The others are because it's actually hassle that people from outside Korea probably can't appreciate. 
I live an hour away from the nearest store and I work during the hours the post office is open. It's also not easy asking for anything out of the ordinary to Koreans where they can't anticipate most of what you are going to say which makes dealing with the postal staff very tricky. 
It's all stress I don't really need in my life just for a couple of bucks and the joy of making a stranger happy (after I stress about their package arriving).
Sorry.

Cafe de BonBon - Bucheon

This is a brand new board game cafe located in Bucheon. Bucheon is on line 1 (dark blue) of the Seoul subway system.

They are very new so organised play doesn't seem to be set up yet but the owner seems keen.
Organised play isn't that important because you will be pleased to know that they pass the hammer test*.

They are open from 2:00 pm to Midnight, hours which are unsurprising in both Korean and gaming culture.

Their Korean address is 원미구 부일로 483번길 15 2층, Bucheon 420-822

They also have a facebook page, I strongly recomend it for people who are fans of fluffy dogs.

https://www.facebook.com/tongtongbonbon

On a serious note they look to have a lot of enthusiasm and if I lived closer to Bucheon I would be really keen to support this store. Their advertising was really cool compared to a lot of Korean magic stores who announce themselves quite dryly and with a minimalist touch to providing information.

*They have a plastic hammer available for assaulting your friends and loved ones with.