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There is no clip analysis tool available within the default application, however you can create your own clip geoprocessing service and use that within ArcGIS Explorer. See the following links below: Using geoprocessing services in ArcGIS Explorer for Windows Desktop Geoprocessing service example: Clip And Ship

To have OpenCycleMap added to ArcGIS Online would involve asking Esri if that would be possible to do. You could do this via its contact form or by submitting an ArcGIS Idea (which I would vote for). An alternative is that ArcGIS Explorer Desktop supports WMS, so if you could find someone providing a service URL for an OpenCycleMap WMS then this could be ...

Without reading all of your question I will guess the answer! You are sending a complete T-SQL 'SELECT' statement when the QueryFilter WhereClause property is just that - a where clause (what comes after the 'WHERE' keyword.) Create a QueryFilter with a valid WhereClause and populate the SubFields property with the fields you want returned.

You can create an animated theme (layer) in ArcExplorer - KML/KMZ is best supported to do this http://webhelp.esri.com/arcgisexplorer/900/en/time_based_animations.htm You need to create a timeline to have the layer animated. As for graphic create a layered gif or png and add into ArcExplorer as a point feature. For ArcGIS (9.2 onwards) there is the ...

The ArcGIS Data Interoperability document details which data formats are supported by ArcGIS Desktop though its optional Interoperability Extension. There is also a FAQ page and whole section devoted to it.