If Trelane had been looking at the 12th/13th century, he wouldn't have known about Napoleon or Alexander Hamilton, been able to quote a Christopher Marlowe play ("Is this the face that launched a thousand ships..."), or been able to play harpsichord sonatas by 18th-century composer Domenico Scarlatti and a Johann Strauss waltz composed in 1880.
All this happened after
Jaeger had speculated that Trelane was viewing the universe through a lightspeed lens.
So Jaeger could have been wholly incorrect in his speculation all along - or then the new evidence fits the speculation perfectly but also helps establish where Trelane's mobile planet has recently been (that is, much closer than 900 ly from Earth).
If you're talking about the pods then the whole ship would have to rotate since the pods don't form a complete circle.
The pods could rotate while the ship was static - the whole point is that there is room there for a full circle (in fact, two full circles), and beyond that, it's just a matter of balancing. But the thing is, we know the pods did not
rotate, yet there was gravity there. And yes, it was there even before our heroes arrived, as Khan wasn't secured in any belts yet hadn't floated away from his mattress, either.
Also, Ceti Alpha =/= Alpha Ceti.
Star Trek is infamous for its use of "cropped" stellar names, such as "Omega" or "Beta". It would make sense that our heroes already know which constellation or corner thereof they are dealing with, and for that reason use shorthand expressions.
Thus, "Ceti Alpha" is in fact something like "Omega Ceti Alpha", in which the final "Alpha" in fact is the phonetic alphabet letter A and indicates the primary component in a wide binary.