If you want group members to add events then you need to edit the group permissions to grant the add and edit event permissions to the members roles. Note it will apply to all calendars int eh group.
There really isn't a way to feed the calendar to other places. The closest is being able to download an ical file for an event to import into supported systems but it is is disconnected(won't reflect changes).
There is Rss Feeds and potenitally REST(though 6.0 calendar REST was still new) but it is custom work