>>I don't know about the "delete after read" option, I didn't know about that and don't know anyone else who ever used it.
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>>That said, with an Exchange account, all messages are on your Exchange server (in the cloud in your case). Everything you use to work with that account is just giving you a view (albeit with some local caching in some cases). If you delete a message in Outlook on a PC, it will soon disappear from your iPhone as well, since the iPhone is just giving you a view of what's in the cloud.
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>>If the "delete after read" option is no longer available, you'll have to delete manually. If you're certain you'll never want to see a message again you can use Shift-Del to bypass the Deleted Items folder.
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>Yes, that is exactly that. I have been showing a demo of that scenario to some people and they all agree this is missing. This is crowding my iPhone with the same inbox I have already read in Outlook and it is a waste of time to go delete all those emails from time to time.

If your iPhone has been set up to access that account as an Exchange mailbox (i.e. Exchange ActiveSync), it should sync automatically - when you delete an Inbox message in Outlook, it also gets deleted from your iPhone. There may be some short delay but it should be pretty quick.

If somehow your iPhone is using another protocol (IMAP or POP3) to access that account, it may not sync that way.

If you have any automatic forwarding set up (with that one or from other accounts) you might need to revisit that. Again, with an Exchange account you're just seeing a view. If you're seeing issues you can delete the account from your iPhone and re-enter it, after a little while it will re-sync.