Is there a setting to display x number of days?
Is there a setting to delete from server?
Pop mail doesn't delete from server or sync mail. So if something is still in the inbox depending on above settings it may display them because they are still on server when push happens.
It's one way. IMAP syncs mail. Personally I like IMAP over POP even if it eats more battery "in theory", but honestly it's negligible.