Usually this is a signal that the sync hasn’t completed yet, especially a first sync on a new device can take a while. Just leaving it sync should normally show the projects jump in place at some point.
The fact you went premium is unrelated to the issue with the categories as sync and categories are unaffected by the license status. If you have everything in iCloud perhaps the best is to simply remove the app from your iPad and install it again from the App Store. It should then also start syncing again and restore things in the proper form. Do make sure to give it a lot of time to complete if it initially seems to again not organise your projects properly in categories as that’s the last stage of the sync process in many cases.
Ah! Yes, that can certainly be off influence, it means that you are basically mixing those projects that you had already in with what is in the cloud. Do you then also see those projects move to the mac? In other words, did the chaos start on your iPad, but now everything is in sync (read: in chaos) on all your devices?