Gary from New Zealand is right; both are correct but each tells a different story about the action "live". The language is a tool for you to use, what story do you want to tell about "live"? Do you want to tell us that at a particular time in the past the action "live" was in progress and not finished? That's good English, or do you want to tell us that the action was completely finished in the past? That is good English too.

Both of your answers are acceptable.

I think that either is fine in those examples. I take it you forgot to delete the 'until' in your second example?

I live in Los Angeles <-- Present Indefinite I am living in Los Angeles <-- Present Continuous