Jabba

It depends on where the clinics are, so I'm guessing it's Gangnam area. If that's the case you can't go wrong with staying near the following stations:
Gangnam Station (Line 1) Yeoksam Station (Line 1) Sinsa Station (Line 3 Apgujeong Station (Line 3) Shin-nonhyeon Station (Line 9) A common thing people do and later regret is staying in Myeongdong or Dongdaemun, it's nice for tourism and shopping, but inconvenient to get to clincis.

You have to factor in staying at least 7 to 8 days after the surgery. So you if you are going to consult and then decide, you might need to factor in consult days, and the buffer time between deciding, and booking what's available. If you'e going to consult 3 to 5 places and then decide at the very least you should budget 13 to 16 days.  Assuming it's not busy or peak season (Nov to Feb). If you are going to decide before you arrive and block out surgery with a deposit. Then you probably can shave it down to 9 to 11 days. 

You'll be able to go out and about with a cast. The first 2 days is tough because there is nasal packing and splints, but this is pulled out the next day or on day 2. 

Starting from Feb 4, 2022 it is 7 days regardless of vaccination status or nationality. 

You can book interpreters through the Gangnam Medical Tour Center, or through Seoul City Office, but if you use these places. The clinic has to be registered with them, sometimes small local clinics are not. Some people have also had some luck using My Seoul Secret. They are an agency, but they also have a plan for people only want to use interpreters and not have them guide or plan their treatments (i.e, get a fee). 
Using in house vs independent interpreters (nice summary)
https://myseoulsecret.com/korean-medical-interpreter-translators/

You can usually scoop some up at airport kiosks when you exit baggage claim, but if you can't do this it's best to get in the city.
You can also pre-oder ones for pick-up at the airport here:
https://www.klook.com/en-US/
I have some information I wrote to a similar question here: 
 

Mango Plate is one that has reviews. 
https://www.mangoplate.com/en/

Here are couple of options:

1) Pre-order SIM with unlimited data plans (BEFORE GOING TO KOREA)

http://www.egsimcard.co.kr/

You have to order this before you arrive to Korea it can be delivered to your accommodation or you can pick it up at the airport. Good for short and long stays.

2) Get SIM / Pocket WiFi at the airport (ALREADY IN KOREA, but at Airport)

There are SIM card and pocket Wifi eggs available for rental. This is convenient for short stays under 15 days.

3) Pre-paid SIM in the City - KT Olleh Shop (Left the airport, but in the city)

https://global.olleh.com:444/eng/contents/about.do

Going to an KT Olleh Global Shop (one near Gangnam Station) will allow you to get a SIM with data plan and a local number. This can be a good option for both short and long stays.

Visit a global shop, it will be easier to deal with.
You need your passport, and must pay in cash only.

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Pocket WiFi's are nice, but having a SIM is just a lot easier, and you don't have to worry about carrying around another device