Married is married. If you marry in South Korea then she's not eligible for a K1 visa. If the consular officer even suspects you've already married then the visa will be denied. If USCIS finds out you married before she came to the US then they'll deny adjustment of status based on the K1 petition. In fact, if this is discovered at any point when dealing with US immigration in the next few years then it will cause problems. It's a can of worms. Don't open it.
Entering the US using the VWP or most other non-immigrant entry passes comes with the condition that you do not intend to become an immigrant before leaving the US. Entering with the preconceived intent to immigrate is illegal. I know some lawyers will recommend you do this because it's possible, and many people get away with it, but that doesn't make it legal, and it also increases the odds the green card application will be denied. Even worse, someone who uses the VWP waives their rights to appeal any decision by an immigration officer. If her adjustment of status were denied then she'd have no option other than to leave the US, potentially with a ban for misrepresentation.
K3 visas are still occasionally issued. The K3 requires the filing of two petitions - an I-130 and an I-129F. If both petitions are approved and sent to the National Visa Center at the same time (and they usually are) then NVC will administratively close the I-129F, and forward only the I-130 to the US consulate. This means that the beneficiary would be applying for an IR1/CR1 visa, regardless if their desire was originally for a K3. The K3 is effectively obsolete.
Get married in Korea. Have a great time. When you return to the US then file an I-130 so your wife can get a CR1 visa.
