Send email from server without creating new Office 365 user account/mailbox

There's no requirement to use a specific FROM address when relaying email through O365 as a specific user. You only need a licensed account with a mailbox.

We've found benefits to relaying our mail though Office 365 which we didn't get from going direct to the receiving mail host:

• You can make use of transport rules, outbound spam, malware filtering, etc.
• If you've configured DNS as guided by Microsoft, you will have set up an SPF record. Annoyingly, (maybe sneakily) the guidelines instruct you to configure an SPF record that uses all 10 of your available SPF entries because of the recursive lookups.
• If it's important to you, you can maintain a certain level of control over who/what is sending email. Just allowing any IP address to email out of your network is risky.

There is also the option to use an internal relay server, optionally with a TLS connection, into a receive connector in Office365. This gets you around the 10k/day recipient limit.

Here's the document you need to read: https://technet.microsoft.com/en-us/library/dn554323(v=exchg.150).aspx

The "From" and "Reply to" addresses, when sent via SMTP, can be literally anything you want them to, as long as you use SMTP authentication from your backend via a single account. I don't believe that O365 will let you configure anonymous relay.

Of course, to send email, you don't need to send through your O365 instance at all. You could just, you know, send the mail. Your backend applications could send mail directly, or you configure an SMTP relay on your network for your backend to use, and then that relay would just send the mail to the intended recipients.

Or as EEAA suggested, you can use a hosted email relay service.