Is it possible to add funds to a US iTunes store account with Paypal?

Yes, you can only use PayPal with iTunes US, and in that PayPal, you'll need to have a US bank account or have your locations and any credit cards/debit's on there will have to be located in the US as iTunes checks for that as well. It all has to be in the US, and cannot have any other bank or CC company for any other country.

I don´t think using PayPal with a Korean bank is possible (haven´t tried this, of course), but if you still want to do this (even though this is probably not legal!), you should look on eBay or similar Sites for iTunes gift cards. Then you could theoretically create a new account in the US Store by trying to buy something that is free and then clicking on "create new account". Only this would enable you to create an account not associated with any credit card. Next you could activate your Gift card and buy from the US store. Note that you would have to lie about having an actual US address, which will make this probably illegal (or at least breaking the terms of service).

As an alternative solution, shop the iTunes US Store whilst in an oversea country with Jerry Cards. Receive high quality scan of the gift cards in your email upon verified payment in your email account, with Paypal or most international credit cards.