Unfortunately, there's not out of box reporting for space admins and writing custom database queries is outside the scope of Jive Customer Support. However, you might utilize the Jive 7 schema (presuming you're using Jive 7.0.2 on-premise) to write a query against the jiveEntitlement table:
Spaces are object type 14. If you require additional assistance writing a query, I would recommend posting the question to the Jive Developers.
As for your second question around reporting on latest activity, you might look into using the CMR "Content Leaderboard" report to see latest activities for a certain timeframe, or use the /content page to filter by latest activity.