Macos – Why can I suddenly not send emails to the contact groups in Outlook

This is not possible.

You could use a rule to forward an email to another account (just mentioning it as an option, even though you mentioned you're not allowed to do this), but you can't write a custom email to a custom person.

I think it would be easier to setup your other mail client to also receive the email you get in outlook.

First, test the email by sending an email to yourself. If you don't receive it back, then you definitely have a problem.

Standard non-Microsoft E-mail typically uses two different protocols: an SMTP server for sending/outgoing, and an IMAP or POP3 server for receiving/incoming.

Sometimes they can be the same server, sometimes they are different. So it may be that your incoming server is incorrect (and different from your outgoing server), or the settings such as Use SSL, etc. are not correct.

Also make sure the incoming mail is using the right protocol, if your provider only supports POP3, IMAP won't work.

If you are using a free email provider you may need to pay them to get POP3/IMAP access, though if you can use their outgoing server without problems, then it's very likely a settings issue.