Why does each user needs their own mailbox?

This is meant to avoid a situation where a few providers end up running mailboxes for everyone, as happened with email. If it doesn't work out then we may need to reconsider this, but we want to try the "one user, one mailbox" approach first and see if it can succeed.

If you have a mailbox and your contacts don't, they can communicate with you via your mailbox. But they can't communicate with each other via your mailbox - for that they would need their own mailboxes.

When I introduce two contacts to each other, do they need to come online at the same time to start using their mailboxes?

Yes, they need to be online at the same time at least once before they can communicate via mailboxes.

The reason for this is that despite all transport properties (Tor, LAN, Bluetooth) being exchanged during introductions, mailbox addresses aren't usual transport properties and are therefore unknown to contacts before them being online at the same time for the first time.

Does the Mailbox owner need to come online with their Briar device to forward messages in group chats via the Mailbox?

Yes, the owner needs to come online with their Briar app in order to forward group chat messages to contacts.