Currently you shout to the group and they might not even know. I think when the owner of a group shouts all the members recieve the shout in there Inbox. This would be helpful to large and small groups.
Support BUT there needs to be a setting in your account page that makes it so you can pick the groups that this featurre comes from and pick a default option for when you join a group
True, This will help if your in a very active clan. Also maybe a setting that if it is only Urgent it gets sent to the mailbox. Plus there could be different settings for each group.