Q: Is it possible to have copies of all emails created in Mail forwarded to my iPhone

What kind of mail account do you have? Please don't post your address or any other account-specific information.

Thanks. That's exactly what I was seeking. Yes, an IMAP account should show you everything, unless you have not selected a folder for synchronization.

For the AOL Account:

Please select Mail->Preferences-> [your AOL account]->Advanced and tell me which check boxes are selected.

Also, have you posed this question to AOL?

Please repeat the steps for the iCloud account.

That's exactly what my accounts show, too. Can you select a folder for one of the accounts and [double tap/right click/whatever] to bring up the context-sensitive menu, then select "Synchronize?"

Excellent - I'm glad to hear that it worked. Please mark this as "solved" so others can benefit.

I see you marked your responses instead of the solution. That's OK.