My experience is absolutely no it does not auto-stop or auto-pause.

Someone from Databricks Sales really should clarify this, because the information in the link above implies that the scaling is automatic.  This expectation of "scaling to zero leaving only the storage cost" has led to unexpected expense.  My instance consumes at least 1 DBU per hour depending on the minimum size setting, and the only control that seems to prevent this is to manually "Stop" the instance.

I'm happy to be proven wrong here.