My experience with other privacy oriented ROMs is that yes you can download apps previously purchased from Google Play via the Aurora app (after providing your Google account credentials to the app) and yes these purchased apps will generally work as expected, barring any underlying issues with the ROM itself not fully supporting parts of the Google API…
However, any apps where you previously made in-app purchases will likely not have those purchased features available. In extreme cases these apps will not even provide basic free features, and might not even launch at all (this basically depends on how the app developer has coded the app to respond to the lack of Play Store in-app purchase verification).
This page from the microG GmsCore Wiki gives status of the microG implementation for various portions of the Google API:
You can see that most of it is working “well enough”, some of it not that well, and some features will never be support. “Such is the price of Freedom!” 