Your Mail app can sort your emails into different categories, but you don't have to use them if you don't want to.

If your iPhone emails aren't updating, several issues could be the culprit. These include incompatible system settings, ...

or another email provider to your phone This article describes how to add email to your iPhone using the built-in Mail app.

One of the longest standing feature requests for Apple Mail on iPhone is for the app to change how it behaves when a user moves a message to a different folder, or sends it to the trash.