Q: Why do messages delete on their own even when they are not set to delete after a certain time period? Why do the messages appear out of order at times?

Hello Shelby,

Thank you for using Apple Support Communities.

I see that you are having a few issues with your Mail app on your iPhone. I would first start with the following steps of restarting the phone and the app:

Restart the app

Close the app and open it again.

Restart your device

Turn your device off and back on.

Check for iOS and app updates

1. Make sure that you're using latest version of iOS.
2. Open the App Store and tap Updates.
3. If updates are available, tap Update All.
4. If asked, enter your Apple ID password. 

You might need a Wi-Fi connection to update some apps.

From this article: If an app you installed unexpectedly quits, stops responding, or won’t open

If these steps do not resolve the issues, then I would recommend restoring your iPhone, that is the only way to reinstall the Mail app should it be at fault. 

Restore your device from an iCloud or iTunes backup

Best regards.