The driving is probably less aggressive in more rural areas, but in Seoul, the drivers deal with 16-lane streets and consistently terrible traffic, so they're definitely dealing with some high nerves. My Lonely Planet tour book said that pedestrian fatality rates in Seoul are much higher than in most major cities. But there is (some level of) order on the streets; it's not as though it's a free-for-all in the way that I've heard it can be in, say, Moscow.

Your plan does indeed sound difficult to accomplish by using mass transit. But do be aware that you'll probably be trying to navigate without much by way of English signage, so make sure you have a good map (itself hard to come by in English, even for Seoul, so likely impossible to find with much detail for rural areas) and an even better Korean phrasebook. And plenty of time and patience.

As far as insurance cover goes, you would probably be covered for any financial penalty (though do check the terms of your policy on international cover; it's not always included). But I think the bigger concern is that you could potentially get called up on some sort of criminal offense. Granted, the chances of anything that dramatic happening are slim, and at that point your embassy would likely get involved, but I have heard that the Korean justice system can be less than completely transparent.

You'll want a mobile phone as well--Koreans do not use the GSM standard as is found elsewhere in the world, so you'll need to rent a phone or buy a new one locally, as your current one won't work (not even on roaming).

And obviously, it would be a good idea to see if you can find some information on Korean traffic laws and practices before you go, so at least you know what the laws are should you run into trouble.