Emails automatically archiving and reappearing after deletion in Apple Mail

2 issues

1. Following the notification that my devices weren't syncing I followed ther advice. But now after 3 or 4 days my emails automatically move to archives. How do I reverse this?

1. When deleting unwanted emails in archives, some of them pop back up again - weird! How to reverse this?

[Re-Titled by Moderator]

Original Title: I rec'd an alert in systems that my devices were not syncing. Having corrected this i can now only see emails from the last 3 or 4 days, the rest are sent to archives.days