If a star is 13000 light years away, doesn't that mean we are seeing 13000-year-old light? If it does, then does that mean when we discover a planet with dimming star light, we are seeing a planet that is 13000 years old? Are we seeing a 13k-year-old orbit?
In one word, yes. Anything and everything we see, we see the way it was a certain time ago—about 1.3 seconds for the Moon, about 13,000 years for your hypothetical planet. Like @Richyt pointed out, there is no way for information to travel faster than the speed of light.
Because any and all celestial objects gravitationally affect any and all others, orbits are not frozen in time and do “evolve.” If this planet is the only “major” thing gravitating around its star, its orbit will change very slowly, but there is still a change, which may or may not be noticeable in 13,000 years, depending on many factors such as the proximity of the planet to its star (the closer it is, the stronger relativistic orbital precession would be—e.g. it is 43″/century for Mercury, 8.6″/century for Venus, and 3.8″/century for the Earth [source for these latter two: https://arxiv.org/pdf/0802.0176.pdf).
To expand on the other answers a bit; if an alien on this exoplanet were to turn his super-powerful telescope in our direction and zoom in on Earth, he would see glaciers and ice-caps reaching down from the North pole to the Alps and bands of hunter-gatherers chasing caribou along the edges of the ice-sheets. Scan about a bit and he might notice some strangely ordered vegetation in the region around Mesopotamia as some of them experimented with agriculture...
Yes and no.
If we are asking if the light has traveled 13,000 years from that planet to our eyes, then yes.
However, if we imply that the light is 13,000 years old in relation to now, the it's not so simple, because the way we look at these things usually presupposes a concept of simultaneity which we are used to, but does not really exist.
According to Einstein's special theory of relativity, it is impossible to say in an absolute sense that two distinct events occur at the same time if those events are separated in space.
If systems move with different speeds, the concept of simultaneity does not make sense any more because observers in these systems do not see things happen in the same way. For example, they will measure different time period between two super novae.
Considering this, we have to accept that the light might have traveled 13,000 light years from the distant star, but observers on the distant planet may not have experienced the same 13,000 years since then like we did.
As far as I can see, yes you would be. There is no way to know what is happening there now, as The theory of relativity would not allow for the propagation of information at faster (or slower, as light travels at the speed of light) than the speed of light, thus 13000 years in this case.
Yes, but only because it is within our galaxy.
For objects which are outside our galaxy, which are not gravitationally bound to us, it is not necessarily true. There are a few twists.
Gravitational lensing is one. Light does not need to take a direct route to reach us. Massive objects will bend light around them. Sometimes this behaves like a lens to sharpen the image, very useful. This can also cause optical illusions such as an Einstein cross making a single image appear four times. Lensing increases the distance light has to travel which can result in the distance traveled (and thus the time) being greater than the straight line distance.
One dramatic illustration of this is the extra-galactic Refsdal Supernova which was predicted to appear in the sky. How? Because they saw it already. The supernova was spotted in Nov 2014. Scientists calculated due to lensing it would reappear sometime in late 2015, and it did in December 2015. Veritasium has a good video on Gravitational Lensing and this prediction.
But that only added a few years, possibly a few decades, to an already multi-billion year journey. What matters far more is redshift. Because the universe is expanding, galaxies are generally moving away from each other. The further away from us, the faster its moving away. The furthest objects are moving away at greater than the speed of light meaning we will never see nor interact with them; what we can still see is the "observable universe".
Distant galaxies are essentially on a conveyor belt moving away from us at a significant fraction of the speed of light. Light from very, very distant galaxies takes much, much longer to reach us than the event. The Refsdal Supernova is 9 billion years old, yet it is 14 billion light-years away. This would seem to mean light is traveling faster than the speed of light, how did it cover 14 billion light-years in 9 billion years?
That's because in those 9 billion years the supernova's galaxy and our own have moved away from each other by over 5 billion light-years. Or more properly, more than 5 billion light-years of space has been added between us and the supernova.
This is how, despite the universe being estimated to be about 13.8 billion years old, we can theoretically see out to about 45 billion light-years. Light from the "edge" started traveling towards us 13.8 billion years ago when everything was much closer together, but the expanding universe caused it to take 13.8 billion years to reach us. Now, due to expansion, those objects are 45 billion light-years away. That 45 billion radius sphere is the Observable Universe.
This does not affect objects within our galaxy because we are gravitationally bound together. As the universe expands, our galaxy does not; gravity holds it together.