Have you actually read the terms of service that you agree to when you sign up for an e-mail account from Microsoft? Obviously not. Because if you did, you'd see that you agreed to let them look at pretty much anything that is sent/received/stored on any Microsoft service for the purposes of detecting and reporting: anything illegal, activity that exploits/harms/threatens to harm children, spam, anything that they (in their sole an infinite wisdom) determine to be inappropriate, engage in fraud, accessing Microsoft's or anyone else's services illegally, copyright infringement, and transmitting malware.
First off: Microsoft is not the government. They can search their own property (their servers) at any time for any reason. 4th amendment does not apply
Second: Even if Microsoft was the government (*shudder*), you gave consent for them to search your communications when you signed up. If the police show up at your house and ask "can we search your house?", if you say yes, then anything they find can be used against you...warrant or not.
Third: The indemnification you talk about only applies to copyright. Any other illegal activities that are perpetuated through their services they can be liable for if they are complacent in them happening on their services.
If you don't like Microsoft searching your e-mails for possible illegal content, don't use Microsoft's e-mail services. That is literally your only recourse. I agree with you in that i'd prefer that no one but me reads my e-mails. This is one of the many reasons why I run my own e-mail server. But Microsoft is within their rights to search your e-mails.