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Build your application with the ArcGIS Android SDK - http://resources.arcgis.com/content/arcgis-android/sdk Requires Android version 2.2 (Froyo) and above. The ArcGIS app will only run on an Android emulator running the 4.0.3 system image with hardware accelaratin enabled. The ArcGIS for Android application has been designed for Android ...

You could use the "Collector for Android" app - http://blogs.esri.com/esri/arcgis/2013/01/16/operations-dashboard-and-collector-for-arcgis-released/ Or write your own application using the ArcGIS Runtime SDK for Android - http://resources.arcgis.com/en/communities/runtime-android/ In either case, you wouldn't need ArcGIS for Server, but could instead use ...

If you don't mind a 3rd party app, you could try Arc2Earth Sync It will connect to Fusion Tables on your account's behalf and make it look like an ArcGIS Feature service. It can also use your ArcGIS.com credentials to create/update web maps so your FT tables show up automatically in the ArcGIS iOS/Android apps (or any of the ArcGIS Online apps/apis). Check ...

MapView.getSpatialReference().getUnit() returns the units of the MapView, which the polygon on the map inherits. You can write a switch statement to handle the conversion to meters from the various units, or perhaps the GeometryEngine.project method will help.

You cannot add data from a shapefile directly to the MapView. You can only add various types of Layers to the Map View. See this Page for more details: Map layer types You could however create a Map Tile Package from a mapservice and use it instead if you desire offline access, by using it in a Local Tile Layer

I'm not overly familiar with the Android API but I am wondering if GeometryEngine.geodesicLength might be an easier approach. GeometryEngine.geodesicLength javadoc