We need to integrate Office 365 with an existing system. For this, we will need to use Office 365 APIs to fetch important resources from Office 365, fetch/download logs for
I don't believe that there are APIs that will capture all of what you want to do. In case you haven't already, you should see if the Office 365 Reporting web service provides any of the information you need. Otherwise, I think you'd need to use PowerShell to connect to the service, then run some scripts for exporting logs, and do this as some scheduled task. That's what I'd look into.