I'm not familiar with all the source systems you've mentioned but Azure Event Hubs is a good service to gather all the logs into one location. If the sources already support exporting logs to Azure Event Hubs, it will make things a lot simpler.
For others, you could ingest logs using Azure Functions or Azure Logic Apps (if a connector exists).
As for processing the logs, either Azure Functions or Azure Logic Apps would do depending on how complex the processing is and the same could be used for sending out alerts as required.
For Logic Apps, you could use the Outlook 365 connector to send out an email for example.