I have now run a troubleshoot of the domain in Office 365 despite the face that is not 72 hours since MX-record have been added (2-3 hours). Status for the domain is and have been "Confirmed" for a while now; is it not supposed to be "Active"?
The troubleshoot gave me an error on the MX-record, and I suspect that is where the problem is.
The CNAME and TXT record is added on the domain controller (the one hosting our domain), but the MX record is added at a company that scans our inbound mails for virus. They have not added a DNS-record but instead - as far as I understood -, they pointed the mails directly at xxxxxxxxxx.mail.eo.outlook.com instead of the IP of our Exchange-server on premise. Is that wrong??