Is a cardinal cubic b-spline just a collection of cubic polynomials (each polynomial connecting two points, and each polynomial defined by those two points and their 2 neighbours)?
If so, then can i express a cardinal cubic b-spline as a collection of cubic b-splines, each defined by four control points?
Apparently the first and fourth control points correspond to the two points being connected. But how do i calculate the coordinates of the other two control points (taking into account the tension value). Also, what happens at the endpoints of the curve?
Yes. Actually any cubic spline is just a string of polynomial cubic segments joined end-to-end. This latter is called the "piecewise polynomial" representation of the spline. Any spline can be represented in piecewise polynomial form.
Yes, you can take each of the polynomial cubic segments mentioned above, and express it in any basis you like. A polynomial cubic segment represented by four b-spline control points (with the knot sequence 0,0,0,0,1,1,1,1) is called a Bezier curve.
Correct.
Since you know the polynomials, you can calculate first derivative $U$ at the start-point $P_0$ and the first derivative $V$ at the end-point $P_3$. The two "interior" control points are then $P_1 = P_0 + \tfrac13U$ and $P_2 = P_3 - \tfrac13V$. A spline segment represented by end-points and end derivatives in this way is said to be in "Hermite" form.