This is a partial answer.

The (minimal smooth compactification of the) surface 
$$X: \quad x^2 + y^2 = f(z)$$
you have written down is an example of a Châtelet surface. These have been studied in great detail by Colliot-Thélène and his collaborators. The key paper relevant to your question is:

Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc and Peter Swinnerton-Dyer - Variétés Stablement Rationnelles Non Rationnelles, Annals of Mathematics.

In particular, in this paper they show that such surfaces are stably rational, unirational, but not rational (provided it has a rational point). So (6) does not hold, but (2) and (5) hold (contrary to what you claim).

They prove this by showing that any *universal torsor* 
$$T \to X$$
with a rational point is rational. Here the universal torsor is a torsor under the Néron-Severi torus. In particular, its generic fibre is geometrically integral, so this is an example where (3) holds.

I don't know in general whether every variety which satisfies (3) must be stably rational; its an interesting question. In the above paper they prove a result which says that this holds for surfaces iff the Picard group of every universal torsor is a stably a permutation Galois module. But it is quite possible that this condition always holds for surfaces. In any case, property (3) seems closely related in general to the rationality of universal torsors.