To put it simple: Director does not offer support for Cult3D files.
It allows the usage of Windows' ActiveX Control, tough, and thus you can run any ActiveX control the user has installed in their computer. So, you can run the Could3D web plugin inside a Director application, since it is an ActiveX control.
The C3D player will have no special privileges or treatment, nor there is any specific lingo commands to handle them: you just control it as you'd control any other ActiveX control, so I advice you to take some reading on interacting with activeX controls in Director.
Using ActiveX controls, fortunally, is rather easy and straightfoward. So much I use the windows' own scrollbars, radio groups, listboxes and etc for some of my applications. But be aware that not all features of the control might be avaliable to Director sometimes.
To use an ActiveX control, go to insert>control>ActiveX, then pick the control you want from the list, and a window with 3 tabs will appear. It contains the control's properties, methods and events.
Properties are the controls' parameters. Be aware that often not all avaliable properties are listed there (i.e.: the Windows Media Player control), you might need to refer to the control doentation to find them all. To modify a property, simple do this, assuming sprite(1) is your ActiveX control:
sprite(1).filename = "sample.mp3"
(as example as if you were using a media player control)
Methods are the control's functions. Stuff like play(), pause(), as example. You call them like this:
Methods often have parameters you pass them, and also can return values. These are listed in the control's properties window and in the doentation.
Events are handlers that are called by the control. All you need to catch one of those is have a lingo handler in a movie script with the same name. As example, the scrollbar activeX has an event called "scroll". If you create a movie script with the following code:
alert "the scrollbar has been moved!"
You'll get a pop up alert everytime you move the scrollbar.