If you are searching on a way to automatically backup gmail on a third party web email provider, stay away from GMX.com
Not only is there no default setting for https but its “mail collector” only gets mail from gmail’s INBOX ONLY. To add insult, they recommend that email in remote emails accounts should be moved into the inbox so that their mail collector works properly.
Q. Can messages be retrieved from subfolders of my third party account?
A. Mail Collector only retrieves messages from the inbox of your third party account. If necessary, please deactivate any filter rules for sorting messages in those accounts, so that all incoming messages will remain in your third party account’s inbox.
To delete your GMX account, follow the setting from here.