One mailbox with multiple additional mailboxes. When sending AS the other mailboxes sent mail should come in those mailboxes sent items. Same for Deleting Items

Use Cache mode otherwise mails will stay in your Outbox.

Method 1: Set the DelegateSentItemsStyle registry value on the Outlook client

Note Outlook must be configured to run in cached mode for this option to work correctly. For more information, see the following Microsoft Knowledge Base article:
 

2703723 Email remains in the Outbox when you use the DelegateSentItemsStyle registry value

If you're running Outlook 2010, install the Outlook 2010 hotfix package that's dated December 14, 2010. Then, follow these steps. 
For more information about this hotfix package, see the following Microsoft Knowledge Base article:

2459115 Description of the Outlook 2010 hotfix package (outlook-x-none.msp): December 14, 2010

Note If you're running Outlook 2013 or a later version, you don't have to install any hotfix.
Click Start, click Run, type regedit, and then click OK.

1. Locate and then click the following registry subkey:

   HKEY_CURRENT_USER\Software\Microsoft\Office\x.0\Outlook\Preferences

   Note The x.0 placeholder represents your version of Office (16.0 = Office 2016, 15.0 = Office 2013, 14.0 = Office 2010).
2. On the Edit menu, point to New, and then click DWORD Value.
3. Type DelegateSentItemsStyle, and then press Enter.
4. Right-click DelegateSentItemsStyle, and then click Modify.
5. In the Value data box, type 1, and then click OK.
6. Exit Registry Editor.

If you want the same to happen when you delete an item from a additional mailbox do this.

To change the registry setting, follow these steps:

1. Quit Outlook.
2. Start Registry Editor. To do this, use one of the following procedures, as appropriate for your version of Windows.
   • Windows 8 and Windows 10: Press Windows Key + R to open a Run dialog box. Type regedit.exeand then press OK.
   • Windows 7: Click Start, type regedit.exein the search box, and then press Enter.
3. Locate the registry key that is appropriate for your version of Outlook.

   HKEY_CURRENT_USER\Software\Microsoft\Office\x.0\Outlook\Options\General

   NoteThe x.0 placeholder represents your version of Office (16.0 = Office 2016, 15.0 = Office 2013, 14.0 = Office 2010, 12.0 = Office 2007).
4. Right-click the DelegateWastebasketStyle value, and then click Modify.
   
   If the key is not present, use the following steps to create it:
   1. Right-click the General folder in the path that is defined in step 4 in the "To Switch the Destination of Deleted Items" section earlier in this article.
   2. Point to New, and then click DWORD Value.
   3. Type DelegateWastebasketStyle, and then press Enter.
5. Change the value data in the Edit DWORD Value dialog box to one of the following values:
   • 8 = Stores deleted items in your folder.
   • 4 = Stores deleted items in the mailbox owner's folder.
   Note Make sure that the delegate user has at least Author level rights for the Deleted Items folder of the owner's mailbox. If the delegate does not have these rights, and this registry option is set to 4, then either the item is deleted permanently or the user receives one of the following error messages:
   • The item could not be deleted, it was either moved or already deleted, or access was denied.
   • Operation Failed.
   • Some items cannot be deleted. They were either moved or already deleted, or access was denied.
   • Could not complete the deletion. The items may have been already deleted or moved.
6. Quit Registry Editor.
7. Restart Outlook.