Exchange issue on domain

Hi All,
Hope you are having a good day!! I am having issues with a new domain that I have recently added to the server.

The setup of the domain is that it has an exchange mailbox with an exterior provider, and on the domain DNS it points the A record to my server so users can access the website, this all works fine and doing what it should be.

The issue is that any other domain that is on the server is unable to send an email to this new domain, can anyone suggest what might be causing this issue?

Also on the website there is a php mail script, which I think may of cause a rate limit on gmail for the whole IP address (server) Is there a why to setup a SPF record or DKIM record when the mail part of the domain is done on another exterior server?

Thank you for looking and helping in advance, it is much appreciated.

Neil