problem 1:

If I "save my sent items" on the server, I get duplicate sent items since I believe the SMTP server saves its own copy and then iMap from the client Mac Mail saves its own. If I disable "save sent items on the server", I now will have 2 folders in the client. One that is really what is sent and one that is synced with the server. FInally, I set it to "Not Save" and do a clean-up every 24 hours on the local sent items.

Problem 2:

If I delete an item from the inbox and move it to the trash, it only deletes it from the inbox when I actually remove it form trash or when I exit Mail but I like to keep mail running in the background. Sometimes I access the mailbox from the web at work where iMap is blocked. From my iPhone through EAS and at home through Mac Mail. So direct delete from inbox is great. I currently have it set to clean up my trash every 24 hours to help with synchronization. 

Enclosed is my setting.