How do I reconfigure local Exchange Server for subdomain when root domain is outsourced?

I have been running a local mail server for the company, but I just switched to an outside mail service provider to handle user emails.

Nevertheless, I still need to run the local Exchange Server 2000 for some administrative purposes.

Firstly, php scripts need to send mails using mail() function, so they need a local smtp server.
Emails sent from php scripts seem to work wihtout any change, but the problem is, mails now end up in the recepients' spam folder. I guess the recepient's mail server don't think mails from my Excahnge Server are legit any more.

Secondly, I need to receive an email to the local Exchange Server so I can programmatically process the email. The email to use would be autoproc@sys.mydomain.com (it used to be autoproc@mydomain.com). Note that I put the subdomain "sys" in the mail address so this particular email would reach my local Exchange Server instead of the outsourced mail server.

And lastly, some softwares such as a 3rd party backup software are configured to send automatic emails to webmaster@mydomain.com using smtp server set as mail.mydomain.com (the local Exchange Server) Now the auto-generated mails should reach email address run by outside service.

I've already updated DNS's MX record so that sys.mydomain.com points to local Exchange Server.

The bottom line question would be "how to reconfigure local Exchange Server for admin purposes (form mail, receive mails to subdomain sys.mydomain.com, etc) when regular user emails are outsourced?"

Thank you.