Will e-mails get deleted automatically?

No, of course not.

As long as you have an account with us, we will not delete any e-mails from you!

The only exception is the trash can, if you set it to this way, and only then.

Note:
Please note that some e-mail programs have settings for deleting e-mails above a certain size, number or age. Therefore, always check the settings of your client before you configure it to use your mailbox.org account or work directly in the mailbox.org web interface.