I moved the .world file out the folder and on to my desktop when I noticed it wasn't gone, that's how I found out it wouldn't make a new one. I don't want to delete it completely because if I can't get a new one to spawn I'd rather have a broken centre and the world than nothing at all.
Back when University Life wasn't working I had to do a factory reset then. Before the factory reset Aurora Skies was working fine and I had the walker and the playpen. After the factory reset I reinstalled all my store content and the walker and play pen never came back.
...Caches. Right. I always forget about those. Hang on a second.
Okay. I uninstalled Aurora Skies and the Centre, closed the Launcher, removed .world file from Sims 3 folder as it didn't delete, removed all main caches, deleted everything in DCBackup except for ccmerged, deleted the Aurora Skies cache files, downloaded neighbourhood from purchase history, reinstalled the neighbourhood, and booting up the game I see: ...No Aurora Skies. The neighbourhood isn't in the drop down menu. That's weird, but it's consistent with what's been happening. Let's try a factory reset folder then.
Renamed The Sims 3 folder to The Sims 3.1, booted up game to create folder, closed game, downloaded Aurora Skies from my purchase history again, installed Aurora Skies, booting up the game I see: Aurora Skies is in the drop down menu! All right, now to go into the neighbourhood to see if the items are working properly. Verdict: Yup! There they are.
Now, I wonder if I could just copy the .world file into my old Sims 3 folder so I don't have to reinstall the rest of the store content. Hrm. I'll have to test this.
Thank you very much for your help Ricalynn!