Similar ESRI arcgis.com can anyone give pointers on how to move layer up/down? And of possible include the context menu.
You can use the map's reorderLayer() method.
The client-side APIs treat each map service as a layer, while the map service itself may also contain layers (as defined in your MXD, for example). If you are asking about layers within an ArcGIS Server map service, it cannot be done as AGS does not support it.