Construction and destruction of Geometries is done using geos::GeometryFactory.
You'll feed it geos::CoordinateSequence for base geometries or vectors of geometries for collections.
If you need to construct geometric shaped geometries, you can use geos::GeometricShapeFactory.
GEOS version info (as a string) can be obtained using geos::geosversion(). The JTS version this release has been ported from is available throu geos::jtsport().
Other standard exceptions are not mapped to this handler, nor does GEOSException inerit from standard exception, so you'll need to catch both if you care (this might change in the future)