BuddyPress.org

You could take a slightly different approach and hide the content but keep the tab: https://buddypress.org/community/groups/how-to-and-troubleshooting/forum/topic/hiding-activity-content-for-non-logged-users/

There is also this (I’ve not used it though): https://buddypress.org/community/groups/private-buddypress/activity/

As for the second bit.. which plugin are you using?