That would be correct, except I don't know what happens to you if you can't get out of EZE. Somehow you'd have to be escorted down to the arrivals/immigration and then, enter Argentina, and then pay when you go back.

For LA to DL, just follow the flight connections signs, and you go through a quick x-ray screen and back to the departures area. I don't know who can help you in the departure area for a DL flight if no one is yet there. I don't even know if there is a lounge for people with DL lounge access @ EZE where you might be able to get post-security/exit migration help before the DL gate is manned. You might want to ask that in the DL forum.