The search for planets outside our solar system has been expanding pretty rapidly recently with the data coming back from the Kepler mission, but nobody has managed to detect a planet quite as far away as a billion light years from Earth. Kepler can only detect Earth-like planets that are less distant than 3000 light years away from our solar system, in a very narrow region of our galaxy.
Our galaxy is about 50,000 light years from center to edge (so about 100,000 light years across), and the next nearest large galaxy is Andromeda, sitting about two and a half million light years away from us. While our current observations of the Milky Way lead us to believe that there’s a planet around pretty much every star in our galaxy, we haven’t been able to survey that much of our own galaxy, let alone the stars in Andromeda, which would be exponentially more difficult to observe. The furthest solid detection of an exoplanet is still only about 21,000 light years away.
But you’re absolutely correct - our images of exoplanets are just as out of date as they are distant from us, and we won’t ever be able to get around that limitation unless we can go visiting them so that the light-delay isn't so severe. A planet that we see at 10,000 light years distant from us will be an image that has traveled for 10,000 years.
On a geological timescale, 10,000 years is just a blip of time - the Earth was pretty much in the same shape as it is now, though we humans had made fewer changes to its surface. On a human timescale, 10,000 years has made a big difference. 10,000 years puts us back into the Neolithic era - the end of the Stone Age, around the time when pottery was developing, and we were beginning to cultivate plants for agriculture. So an intelligent civilization, 10,000 light years distant, that is just now looking for other life in the Universe would spy our Earth as a rocky planet with an atmosphere, far enough away from our sun that water could exist in our atmosphere, and if they managed to examine our atmosphere, they would notice that it is mostly nitrogen, with some oxygen and carbon dioxide in it as well, and that it contains water vapor. They would not be able to tell that there are creatures on that planet that are 10,000 years away from developing the internet, neurosurgery, and machines able to detect tiny distortions in space itself.
This kind of time delay is one of the reasons that scientists get extra excited when they find a nearby rocky planet that might be able to have liquid water on its surface - if the planet is close to us, then the time delay isn’t as bad as a more distant planet. (It is also much easier to observe these nearby planets in any degree of detail - the farther away you are from the Earth, the harder these measurements get.) We only managed to detect the contents of the atmosphere of a slightly-bigger-than-Earth planet for the first time a few days ago — unfortunately that planet is totally devoid of water, having an atmosphere of mostly hydrogen and helium, with some hydrogen cyanide thrown in for extra poisonous flavor. This planet is only 60 light years away, so our image of it is only out of date as far as 1976— this particular planet won’t have evolved into a friendlier, life-hosting planet in such a short time.
But let’s say a super-intelligent civilization out there has built an impossibly large telescope, and has the power (and the time) to detect planets orbiting stars in a distant galaxy, and they pointed it at our Earth. If they happened to be 2.5 billion light years distant, our planet’s atmosphere would be in the middle of a dramatic change. 2.5 billion years ago, our planet was in the middle of the Oxygen Catastrophe - the earliest photosynthetic bacteria were dumping oxygen into the atmosphere faster than it could be absorbed, and oxygen was slowly building up. As oxygen was a toxic byproduct to the single-celled life which had been living in a delightfully oxygen-free environment, they would have to adapt or die off. Observations of our planet from that distance would be able only to tell the observer that our planet existed, it has water in its atmosphere, and how rapidly we travel around our star, but not so much as a hint to our space-exploring future.
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