Using Classic Outlook on Windows for personal email, calendar, and contact management
Domain name and DNS records are two different things. DNS records point to a domain name. Even though you do not own hotmail.fr you can still configure dkim if you know what you are doing and have a premium account. If you are unsure how to configure dkim in hotmail then you can reach out to support here: https://support.microsoft.com/en-us/office/how-to-get-support-for-outlook-com-f5482a98-616c-4d44-b7c5-8aaaadf5c11a
This is an article that explains the problem that all mail admins should have been prepared for: https://blogs.cisco.com/security/february-1-2024-a-date-all-email-senders-should-care-about and here https://www.proofpoint.com/us/blog/email-and-cloud-threats/google-and-yahoo-set-new-email-authentication-requirements
Not many mail admins use hotmail. But for people who do, I said you need to test with tools like mx****. Hotmail already has the dns records setup for you if you use the free service and the free service does not offer dmarc or dkim signing. You have to have a premium plan or office 365 subscription as I mentioned.
WIth office 365 you can setup dkim here: https://learn.microsoft.com/en-us/microsoft-365/security/office-365-security/email-authentication-dkim-configure?view=o365-worldwide
Good luck