The core logs can be piped to a file / process using which you can send the logs to Datadog. We have core configs using which you can provide file paths for where logging should happen, and if set to null, it will log to the console.
From the backend SDK, you can override all the recipe functions for the recipe you are using and add a call to your logging service in your override before calling the original implement on.
In terms of liveness check, you can use the /hello API on the core - if that returns a 200, the core and the associated db are running fine.