Think of this as a mirror. If you read, delete, or move an email in Thunderbird, the same action is instantly mirrored on the server and on all your other devices (like your phone). Does it leave messages on the server? Yes, by default.
Note: If you are using IMAP, this option is not available because leaving messages on the server is the default behavior. does thunderbird download messages from server
With POP3, Thunderbird downloads messages and, . Think of this as a mirror
Tick the box: .
Emails are stored locally, making them available without internet. or move an email in Thunderbird
Not unless you change settings.