This container you are referring to is supposed to store DataViews in it (please take a look at the getDefaultDataViews method from the BaseDataView module). It is not supposed to store buttons in it.
It is better to add the button to a separate container created in the FiltersContainer container and bind process execution to this button. As an example I've added the button to the contact section analytics data view:
For the first question: I used the FiltersContainer container as a parent for the button, but it has the logic that it's hidden in the list view and is displayed only in analytics view (logic of the saveFiltersContainersVisibility method). In this case logic like this should be used: