Hello,

this is in some way in between a problem and an idea. 

When sharing folders (which is actually not any close to a shared mailbox...) the person which the folder has been shared with can access that folder and can send emails when delegation is set. Okay, so far. But there is one major issue. When this person is sending an email from the shared address, this email is in its own sent items. Furthermore, is is not possible to move it to the shared folder. If that would be possible, you could use filters. 

Is there a way to get around this?

By the way: This has been said many times - TeamInbox is not useful at all when not integrated with mail.