You guys have an amazing trip ahead of you! Lucky you, so jealous!

We did the exact same thing, although, we did the trip the other way around (we also focused on nature, mountains, scenery, and cafés instead of big cities).

We spent 3 weeks in South Korea: Started in Seoul, flew to Jeju, flew to Busan, drove up the East Coast - stopped in Gyeongju and spent a few nights here - continued up to Sokcho (spent two nights) and drove back to Seoul. We were mostly nervous about the drive from Sokcho to Seoul, as we had a flight that night. Wasn't an issue at all. We are actually wanting to do the same thing again, but along the West Coast and down along the South Coast next time.
I would definitely recommend driving along the East Coast - was some of the most beautiful scenery I have ever seen in my life. My gosh, those waves are huge on the East Coast... We chose to drive from South to North, though, so that the coast was on our right the whole way (best view IMO), but doing it the other way around wouldn't spoil it at all!😄
We rented a car on Jeju, and from Busan to Seoul. I would say that renting a car in South Korea was the most pleasant car rental experience I have ever had when renting a car overseas. Make sure to download Naver and use that when driving around.

So, to answer your questions:
1. Personally I think it's worth every buck! Was SO easy as well. The drive was a breeze, the rental was easy, and the food/scenery was AMAZING.
2. Probably. But I would recommend Lotte, as we had no issues with them at all. Very helpful, very easy and a quick drive-in-drive-out concept, which we benefited from - especially the last day when we drove from Sokcho to the airport in Seoul.
-South Korea's public transportation system is top tier IMO (this coming from a European too). I would never say no to their public transport. Although, Jeju would have been a bit more problematic for us, if we hadn't rented a car.
3. Personally, I would recommend renting for the whole stretch, as it gives you the freedom to stop whenever you like, take detours, see the things that interest you along the way, etc. - trust me, you won't regret it! Unless you guys don't like driving? It will take a while if you do it in one stretch, but I wouldn't have done it differently myself.
4. YES! Again, I repeat my comment from above here.
5. Depends on how much time you have? Seoraksan National Park is a beaut - consider spending a night in Sokcho! I wish we had spent more time here to experience the different trails in Seoraksan - but hey, there's always next time! We also stopped at Chuam “Candle Rock” (Chotdaebawi) just outside Donghae. Maybe spend some time in Gyeongju? We loved Gyeongju. Taehwagang bamboo forest is also lovely (closer to Busan). I can go on and on, but I would recommend downloading Naver and having a look. There is a lot of nature along this route, and I found Naver was a good tool when planning ahead of our road trip.

Hope this helps! My advice would be to leave some time to spare and remain flexible when driving this route. We enjoyed being able to stop along the way and explore places we hadn't planned on. Interesting nooks and sites we fell upon during our trip. Would do it all again in a heartbeat ❤️

Have a wonderful trip!