Hi Roche,
1). You can figure out the appropriate class consulting the API of any particular method you would like to use. Most often they work on TopoDS_Shape, sometimes directly on geometrical objects - Handle(Geom_Curve), Handle(Geom_Surface), etc.
2). Adaptor classes implement an "adaptor" pattern (if you are familiar with such) wrapping different representations into some common interface. For example, there is an adaptor to represent curve (Adaptor3d_Curve) from really a curve (GeomAdaptor_Curve) or edge (BRepAdaptor_Edge) or even wire (BRepAdaptor_CompCurve).
If documentation does shed enough light, CDL and source code should be quite self-explaining.
Hope this helps.
Commenter-1