Outlook Mac – Move Email Messages Between Different Accounts

From what I read, you actually are almost there. It looks like your problem may be how the folders are done (as in, locally or on the server).

As you said in your answer, you copy the emails to local folders, so therefore they would never be on your phone. And with iCloud's limited rules (in the cloud), as you note, you cant really make a sound or do much with them either.

As long as your Mac is always running (to run the rules), you really would only need to update your local mail rules to put the emails in the servers folders. This way if you are at your computer, you would hear the audio cues, as well as have the mail filtered. If you step away, as long as your Mac could still process the mail, it will copy the emails to the correct server folders, and be available.

On your Mac now, you should see two inboxes, iCloud and 'On My Mac'. Make sure folders exist under the iCloud account for where these messages should go. Then update your rules to point to the iCloud account folders, versus those on your mac.

The only thing you may see from this is some messages will be in your inbox for a very short period of time (before your Mac runs its rules), versus iClouds server side rules that process mail before its delivered.

This should happen automatically and no special setup is required. Click on the email account's inbox (on the left side of the window) that you would like to see just that inbox. This should pull up the inbox for that account only.