Domestic flight for departure: 20 mins before boarding. I never do the transit coz I visit Seoul first then only Jeju. Better make 1 hour earlier than boarding time.

Apps you MUST download during your trips:
For Seoul: Subway Korea (it shows you how many station from your location to your destination - Airport station include as well)
For Busan: Busan Metro
Both has the entire route of the subway. It is extremely useful.

Jeju: I just use google map and their GPS. That's already sufficient.

Rent and return car in Jeju is very convenient. I just return it and after car inspection less than 5 minutes, I can go already. Then just hop on car rental shuttle bus to airport.
Car rental office to airport roughly 10 minutes driving distance.
If return car later than 30 minutes or an hour from the time you mention..that's alright. They don't bother much.

There's only "Book A Car" or "Jeju Book A Car" option on the page. Either one will be good.
Well, there's no English navigation. It is still a Korean Navigation where you can choose English/Korean Speaking. The lotte guy help to change to English speaking but later not sure why it speak in Korean again.
I have a hard time to change back to English speaking, in the end I just give up and let it speak in Korean.

Just follow the navigation can already. Coz their GPS is super accurate and no laggy etc. You won't miss the turning.
As long as you provide the destination phone number..you are good to go.

For my experience, I went to East and West from Jeju-Si. Ya, I stay at Jeju-Si only.
The travel time from here to East/West each about 1.5 hour or less.

Jeju-Si has more shops and stalls open during night time while Seogwipo is already silence around 8pm.
I didn't go to Namwon-eup, so can't comment it ya.

After all it still depends on which place you gonna visit. Mind to tell your itinerary at Jeju?

Even though below show from Jeju-Si to East/West so far away from each other...actually driving time is around 1.5hour only.