There is no contradiction in our world and a distant world BOTH seeing each other's past. Since we are both emitting photons (radio waves, light waves, heat, etc.) at the same speed, these photons would take the same amount of time to cross the expanse whether they were going from here to there or there to here. So if we were both in the year 1999 and we were 1000 light years away, then we would both see what conditions were like for the other world in the year 999. Of course, there weren't a heck of a lot of emissions coming from our planet in 999 that could be distinguished from forest fires, lightning storms, etc. Maybe this is why we haven't had any certain evidence of visitors from other worlds: we haven't been around long enough for our nocturnal emissions to reach them and send them on their way in our direction!
Seeing the past of another world may be interesting, but it will not tell us much about ourselves, at least not directly. If we learn from the transmissions about technologies that we have yet to discover, then of course this would be very helpful to us. If, for example, one of the news reports suggests an interesting alloy of metal that is lighter and stronger than anything we have developed to date, and gives enough details for us to create it on earth, then that would help us move forward into the future a bit faster. The only way, however, that we could more clearly discover our past is if some other world has been making the equivalent of "home movies" while watching our Earth's past then somehow communicates this to us. If they are watching our history of 1000 years ago, then we will have to wait another 1000 years for the retransmissions to reach us, but we might be quite interested nevertheless.
Your second question is much more interesting. As our world careens through space we are constantly emitting photons. These photons travel much faster than the Earth. Is it possible for the path of the Earth to pass through the photons emitted by earlier transmissions from earth? I don't have the mental wherewithal right now to sit down and try to figure this out, but I think it would be worth considering. It might even be something that could get funded as an experiment to send aboard one of the future Space Shuttle flights. Of course, the military might not like it as some of the earlier transmissions were messages encoded with far simpler encryption algorithms that could easily be decrypted with technology available to the average PC user. Imagine if we could prove once and for all the FDR knew the Japanese were going to bomb Pearl Harbor and did nothing about it. That might ruin his chances of ending up as Time's Person of the Century!!