So, this may be of limited use, but to use JFK-MEX as a guide (I don't see why it would be different?)

1) If it is an AM plane, AM does the bag drop. If it is a Delta codeshare, go to Delta drop. AM tends to open up counter/drop 3-4 hours before the flight.
2) If you are flying AM business. you should have access to Delta Lounge. https://bit.ly/2SP7nQf

LAX specific, this is what I see (though I have not done it personally)
3) Aerolitoral, Aeromexico, Delta, Virgin Atlantic passengers will check-in at Terminal 2; walk to Terminal B where flights depart/arrive (https://www.flylax.com/-/media/flyla...ation-map.ashx)
Delta lounge is Terminal 2, and Korean Air (Priority Pass) is Terminal B (also known as Tom Bradley International Terminal). So you should be fine in either. By the looks of it, KAL will be closer to your gate so might be the better choice regardless...