When sending mail I get an error message for each email “Error 5xx (we do not relay)” or “authentication failed” or comparable

Such error messages don’t come from the software. BulkMailer shows the exact response from the mail server.

It is likely to SMTP authentication. Most mail servers now require a login. Select Extras->Options->Connection
(x) Smtp server requires authentication
and enter your user name and password to the mailbox specified by the email sender address.