Okay, I gave both Korean Air and United a call. Looks like I should be able to check my bags through Korean Air to the United flight as they have baggage agreements, even though the two flights are on different itinerary/tickets.

Furthermore if I had to retrieve my checked bags and recheck them for my next flight the minimal time I need is 1 hour 40 minutes, which means I should have plenty of time to spare either way.

And I got that the lounge I'm suppose to use is the Asiana Lounge.