BuddyPress

What’s the best way to display recent forum topics and activity in a widget style format?

I don’t know about an off the shelf widget, but you could write one easily enough using the activity loop and filtering for forum posts:

http://codex.buddypress.org/developer-docs/custom-buddypress-loops/the-activity-stream-loop/

ok, you could also look at these plugins by @nuprn1 : http://buddypress.org/community/groups/buddypress-group-forum-extras/