I’ve installed the Salesforce managed package and I am able to start an AIRe Link session from within Salesforce.
Does anyone know what is the best practice for getting the session data into Salesforce? Is there any integration for doing this or do you have any other suggestions?

The SalesForce integration is very basic, i allows synchronization of very limited set of data. It is possible to get more information from AIRe Link, but not within the scope of the package itself. It would be needed to use AIRe Link API directly, probably within some custom integration.