Additionally, Kies will only update if you are one firmware behind.

For Kies to offer an update, your phone firmware CSC ( dial *#1234# to see this) and your phone Sales Code, have to geographically match.
First of all, Samsung recommend that you update with Kies.

Ok, let's look at Kies first.
Your phone has 2 codes that have to match for Kies to update.
1. Product Code - this is the country/network your phone was sold in (as you point out India has 2 codes INU & INS)
2. The firmware CSC code.

To find both of these codes you can install CheckFUS from the Play Store.

If you are happy to update your phone without Kies, then ignore the information above.

Question 1. Do you want to install SER JB firmware?
If you do, then Kies will update in future. 
Install procedure :
 Download the firmware to your PC and Unzip once to get a tar file. 
 Now go to this link on how to install.
To both ansh & pandilica. 
For Kies to update you need 2 things. 
1. The Product Code and Firmware CSC have to match. 
2. The firmware has to be only one update behind. 

Using ansh's firmware as an example. 
Product Code ends with 
INU is a Product Code for India. 
But there are 2 Product Codes and firmware for India. 
They are INU and ODD. 

Ansh has installed the wrong firmware, and that is why Kies will not recognise the phone.
Question 2
Anyway back to the issue at hand.
Your phone should say BTU for the Sales Code (as seen in Kies) and the reason it doesn't is that KOR is the home country code (KOREA)

Now it doesn't really matter, except that Kies will not in fact it hasn't as you are 3 updates behind.
See here:

Now the only way you can update is with a firmware installer called Odin - it's like Kies but much better. 

First job:
1. Download the top firmware on the list to your computer (it takes around 40 minutes)
Unzip once and save the .tar file where you will find it again (ignore the .dll file)
2. While waiting, backup your phone either with Kies or an app from the Play Store called Carbon.
3. Go to Settings > Developer options and select USB debugging
4. Now go to this link and install Odin and read how to install the firmware.
5. Prepare your phone by going to Recovery Mode:
Press and hold Volume up + Menu button + Power button (all 3 at the same time)
6. In the blue writing of the Recovery menu use volume to select Clear cache and power button to action.
7. Now do same for Clear data/Factory reset (this will shut down your phone and it will wipe all data except photographs so do make sure you've backed up, especially contacts)
8. On restart, do not fill in details, but shut down straight away then start in Download Mode:
Same as Recovery Mode except press volume down until in yellow triangle screen. Press volume up to show android figure.
9. Now close Kies and open Odin and put your downloaded file from 1 in the PDA box of Odin.
10. Connect phone with original USB lead and wait for ID:COM box to change colour>
11. Press start - nothing may happen for a few minutes, but eventually your firmware will be installed and Odin will say 'PASS'.
12 Your phone will restart and you will need:
a. Your wifi password
b. Your Google email and password
c. Your Samsung email and password
d. Dropbox email and password (if you have an account - it's a good idea to have one to sync your data/photographs)

That's about it.