The map doesn’t center when an icon is clicked, but it moves to make room for the popup. Both Leaflet and Google work this way by default.
You cannot use parent pages for filtering, but you could assign parents+children to the same taxonomy (categories, tags, or a custom taxonomy) and use that.
If you are familiar with PHP, you could also use filter
mappress_query_filter to adjust the query results based on the filters. Please see file ‘mappress_query’ to see where that filter is called.