Send mail from a shared mailbox - Save-Sent-Items-To - Direct replies to

1. Open Outlook.
2. Choose New Email. 
3. If you don't see the From field at the top of your message, choose Options > From.

4. Click From in the message, and change to the shared email address. If you don't see your shared email address, choose Other email address and then type in the shared email address.             

5. then select The Options Tab, Select Direct Replies To:

6. under Delivery Options Select the check box Have replies sent to and Select Names you want the replies sent to, your mailbox or the shared mailbox.  

7. Choose OK.

8. Finish typing your message and then choose Send.

From now on when you want to send a message from your shared mailbox, the address will be available in your From drop down list.

You can also Save a copy of the Sent message to a folder of your choice

When you are composing an email message, you can define an alternative folder where this email will be stored after sending.

1: Click the Save Sent Item To and then Other Folder...   in the Options Tab in the new email message.

2. In the Select Folder dialog box, select and highlight a folder in the Folders: box. Scroll down to the shared mailbox and select the Sent folder of the email you're sending from and click the OK button.

Result:

When multiple people have access to this mailbox they will be able to see all emails being sent and replied to after setting up the above steps.