If an exoplanet transit we are seeing is 13000 light years away, are we seeing a 13000-year-old orbit? 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).