There are various ways to achieve this. I currently have a similar arrangement, albeit with one computer. Here are some options:
Incoming Emails
Option 1
Do you have control of the Exchange server? Some version of Windows Server have a POP3 Connector - I know Windows SBS Server 2008 does. You can configure this to download emails from a POP3 account and place them into an Exchange user's mailbox. This will automatically distribute the incoming email to all copies of Outlook that are connected to that Exchange account.
If your version of Windows Server doesn't include it, you can purchase third party software with similar functionality, eg: http://www.mapilab.com/exchange/exchange_pop3/
Option 2
You can add a POP3 account to Outlook as you would normally. Newer versions of Outlook allow you to choose where this account will deliver incoming emails - so after you have set up the account, under Account Settings, click Change Folder and select the inbox folder for the Exchange account, to deliver the emails to.
You can either set up this POP3 account on each of the three computers, or pick one. If you choose one, that computer will need to stay on with Outlook open so it can download incoming emails. However, if you set up each computer to be able to download POP3 emails, you will need to untick the following setting - 'Leave a copy of messages on the server'. If this stays ticked, each computer will download each email, resulting in three copies of the same email.
Option 3
If you have control over the email server which handles incoming email for your domain, set up a rule on this server which forwards incoming emails to the email address of the Exchange account.
Outgoing Emails
Option 1
If this is available to you, configure the Exchange account to send from the new address. I don't know whether you administer the Exchange Server - if you do, I might be able to provide advice in the comments. Otherwise contact the administrator.
Option 2
Set up a POP3 account on each computer. For any email that the user needs to send from the new address, they can select the account in the From field in Outlook. This will usually have to be done manually, unless you went with Option 2 for the incoming emails. If Outlook downloads the incoming email via POP3, it is clever enough to reply to emails using the same email account.
Note that if you set up a POP3 account on each computer, you may end up with multiple copies of the same emails (see my notes above in Option 2 for Incoming Emails). To avoid this, either enter false details for the incoming mail server, so that it doesn't successfully download any emails, or disable the email account from receiving emails in the Send/Receive settings.
Sorry for the long answer, but without knowing your exact setup, I can't provide more specific advice. I hope this helps.
Yes, I have control of the Exchange server, it's ES2013. – matthewk – 2014-03-11T20:42:16.603
Then you can modify the user's mailbox - set the new email address to be the primary one - http://technet.microsoft.com/en-us/library/bb123794(v=exchg.150).aspx. This will mean that outgoing emails are automatically from the new domain. For incoming emails, either set up your domain hosting to forward them to the exchange account, or use a POP3 connector. Will that work for you?
– Garrulinae – 2014-03-12T00:56:51.503I don't want the new domain to be managed by the Exchange server, because other users are using it via their Gmail accounts, and I want to maintain that setup for them. And I don't want change this user's primary email address. I just want him to be able to choose to use it as necessary, and be able to access email sent to that address or from that address from any of his three computers. – matthewk – 2014-03-12T11:49:45.553
OK, you have three options for incoming emails and one option for outgoing emails, then. Could you accept this answer, please? – Garrulinae – 2014-03-13T01:23:55.020
I've awarded the bounty, but have not accepted the answer because I haven't been able to try them out yet. If one of them works as required, I will also accept the answer. Thanks for your time. – matthewk – 2014-03-13T18:05:09.677