I have all the addresses already set up. Not a problem there.
You need to set up all instances of your Outlook to leave a copy on the server (your choice of how long) for every account. Once the mail Client has acquired an email it is removing it from the server and none of the other clients will see it.
BTW this leaves you with multiple copies of the same email across your PC’s. If you reply on one, that history will not be on the others. You may be able to share your .ost and .pst files on the home network, but I think that could be a problem with multiple versions of Outlook.