With some email clients using pop3, you have to manually delete the message on your device, and then again on the server. Some have the option to do both, but you have to have the option set up on the website, and on your device. For your second issue, if you change the option from say keep mail for a certain number of days, to another, it will redownload your emails for the new period of time you set it for. Hope this rambling makes sense.
