How do I prevent emails from disappearing on my MacBook Air after setting up a new email account?

Yesterday I added a MacBook Air/Sequoia (have a Mac desktop and iPhone) and set up my email account there. The Air will not keep emails once they are off the mail server. Emails disappear even though I have 'never' selected as delete option. I want to be able to keep some emails permanently, not have them disappear off the MacBook.

Both my Mac desktop and iPhone used to allow me to keep emails indefinitely, as long as the email wasn't downloaded to the MacBookAir first. But now everything is a mess of disappearing emails.

Additional info - the MacBookAir keeps creating additional folders with similar names, too. For instance, there's a trash with an trashcan icon but also a deleted messages folder with folder icon.

Is this issue with the MacBookAir, my email server or both? How can I fix this so emails flow thru my server to be saved indefinitely on my laptop? Please help.

[Re-Titled by Moderator]