Is the Recommendation rule compatible with 3rd party product recommendation extensions?

it depends.

There are a few different methods which the trigger can use to generate recommendations.

Some of those methods rely upon data stored in Magento (such as Related Product data).

If you're using a 3rd party extension that populates that data for you, then you're all set. MageMail can use that data that the extension is populating.

If the extension uses it's own custom database table or some other data source, then MageMail won't be able to leverage that.