Is it possible to output a Geometry object in the http://geojson.org/ format (as a simple string) somehow? The runtime only seems to support ToJson(), which is a ArcGIS-specific format.
Maybe there is an open source library or a C# code snippet out there that does that?
There's no GeoJson support out of the box. I'm not aware of any open-source .NET libraries, but since GeoJSON is rather simple, I doubt it would take you very long to write a simple serializer
Perhaps this library will give you a headstart? https://github.com/GeoJSON-Net/GeoJSON.Net
There are probably more, but this was the first I found when searching for one.