A light year is a measure of distance, not time. It's how far light travels in the time frame of 1 year, roughly 6 trillion miles. "Light years" is how far away other objects outside our solar system are, not long long it takes light to reach them. How long it takes for light to travel from here to there or there to here is just measured in years.
As far as looking into the past, that is correct. We see the sun as it was 8 minutes ago. We see the other planets as they were many minutes or hours ago. We see the next nearest star as it was 4 years ago. We see the Andromeda galaxy as it looked about 2.5 million years ago. And we see the most distant galaxies as they looked 12 or 13 billion years ago, because that's how long it's taken for their light to reach us. So yes, looking out into space really is looking into a window of the past, with the amount of time looking back into the past depending on how far away the particular object is. In your hypothetical scenario, yes, a huge telescope 2000 light years away would be just now seeing Earth as it was 2000 years ago, although at that distance the best it might be able to see the Earth would be a tiny fuzzy dot being swamped out by the light of the sun.
Another thing to consider: let's say we start receiving radio signals from an alien civilization on the other side of the galaxy, let's say 50,000 light years away. That civilization might not even exist any more due to any number of catastrophic reasons, and we'd never be able to let them know we got the message. When you go outside on a clear night and start thinking about these distances as you look up at the stars, it is just mind blowing to think about.