Having not played Leviathan yet, I'm going to go with what I've seen in the previous games: It doesn't really matter. The best example would be Arrival for ME2, which had slightly different dialog lines depending on if you did it before taking on the Collector base or after. Okay, it mattered in ME1, since there was no end game plus, but it didn't matter in ME2.
But ME3 is a little different because there is no real end game plus (which is different from new game plus), but rather after beating the game, it puts you at the last save before the point of no return. In ME2, you just kept going until you decided to start a new game plus or new game.
For this reason, if you haven't played through already, I would personally do the DLC before completing the game. There are many benefits to doing this as well, such as possibly receiving extra War Assets that do you basically nothing after beating the game (unless you want to do the ending again). But, you are still given the chance to do the DLC after beating the game, so that you don't have to start from scratch.
So, definitely do the expanded ending and don't bother with the original, and I would highly suggest doing Leviathan before beating the game if possible. Based on the wiki, Leviathan won't be available at the very beginning, which is good, because I wouldn't play it very early anyways.