Regarding how engines work, I can't speak for the Cobra engine, but in Unity I can create a CoRoutine that will run in the background on the game loop to do these kinds of tasks. I am assuming (granted, we know what that means) that the Cobra engine has something similar. In terms of stations, off the top of my head, they don't "move" - they just orbit at the same rate as the planet does, so appear static. But I'm not a Frontier dev, and I can't speak for how they do it - it's a black art for me - and kudos to them for getting that part right.
Spend some time experimenting with how space works in game. For example, target a station in the game, fly into its orbital path about 5 mega-meters away from station, then drop into normal space, and set throttle to zero. Observe what happens to the station: depending upon the body its orbiting, it'll either approach you rapidly, or move away from you so rapidly even a high-speed iCourrier wouldn't be able to keep up. If it's approaching you, at one mega-meter, you enter its "sphere of influence*" (assuming everything works correctly, of course) and you'll come to a complete stop relative to the station.
For bonus points, observe what happens to other POIs in the system, especially on the (landable) planet the station is orbiting, once you're in the station's SOI. For even more extra credit, pay attention to system POIs when you're in a planetary ring.
Everything orbits something, and its the job of the game to keep track of that motion.
I can only think of five games that have attempted this level of detail: Fronter: Elite 2, Frontier First Encounters, Elite: Dangerous, Kerbal Space Program, and Star Citizen. Three are by Frontier Developments, one simulates a single solar system, and one is an utter disaster.
*Spheres of Influence date back to Frontier: Elite 2, and are one of those necessary compromises required to keep things simple enough that a home computer will have enough processing power to do everything it needs to do. SOI changes are also were the illusion created by Frontier tends to break. Frontier did such a good job disguising SOI changes during normal use that you have to jump through hoops to see them in action. SOI changes are also why I believe Frontier artificially requires you to throttle down, despite the FSS being perfectly usable at high speeds: SOI changes are
extremly noticable in the FSS for some reason, sometimes jarringly so.
Curiously, Kerbal Space Program also uses SOIs as part of their simulation. I don't know what system Star Citizen uses, but it wouldn't surprise me at all if they're doing their own thing, rather than goes with what obviously works in other games. The KSP Devs back before the game went live talked quite a bit about how difficult it was to get everything working correctly, and
they had just a single, and simple, solar system. Frontier has to make sure it works for 400 billion systems, some of which have up to nine nested orbital hierarchies.