They are all called 'control points' because they all influence the shape of the Bezier curve. The first and the last points are only different because they define the start and end position of the curve.
The cubic curve is the most common Bezier curve used in computer graphics and has 4 control points, but a Bezier curve can be created from any number of control points (minimum 2). For instance in the sketch Knot in 3D the tube has been created from a single Bezier curve with 26 control points.
Wikipedia demonstartes different order Bezier cuves - about half way down this page
No but if you join all the control points in order (the green line) it is called the hull
The line 0-->1 is the tangent of the curve at point 0 and the line 3-->2 is the tangent of the curve at point 3
So you can call the lines anything you like but the word 'handle' is normally used to identify a small area that can be dragged with the mouse. In this picture the blue/white circles could be 'handles' to drag the control points.