Do purchases carry over between the web, Google Play, and the App Store?

Yes. Pro and storage attach to your account, not to a platform or store. Buy on the web and every app you sign into is Pro, including store builds; buy inside a store build and the web and desktop apps unlock the same way. There is no "restore purchases" button because signing in is the restore - the entitlement follows the account.

The one difference is who bills you. Web purchases run through Paddle and are managed in the app under Settings > Storage. Purchases made inside a store build (Google Play, or the App Store once the iOS app ships) are billed by that store: recurring storage is cancelled in the store's subscription settings, and refunds follow the store's process. Either way it is the same account and the same Pro everywhere - including platforms that do not exist yet.