I agree with you most of the parts. Korea is Internet-Explorer oriented nation, so most websites (especially government running websites) do not work with firefox or safari. If you begin with us.flyasiana.com as I do, instead of flyasiana.com, you'll directly get into Asiana USA website. If you're not using IE, you'll never get what you want. If you're in front of PC with IE, still the website requires to install weird programs called Active-X which you don't want at all. Just keep away from any Korean websites - that's the safest way to keep your PC from all the viruses. If you're using Mac, find PC.
Foreign card is extremely hard to use in Korea or any Korean company website. Because they usually require one stupid verification process which is impossible with foreign card. That online transaction system may work with IE, but not Firefox.
Then how to buy Korean domestic ticket? Orbitz has it, but with some fee - $5 or $10, I guess. Or, can you call Asiana and get some help? All other Korean travel agency websites are even worse than flyasiana.com, so you don't want them either. Sorry, but south Korea is like north Korea for firefox users - non-approachable.