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.

Let $k$ be a field of characteristic $0$, $a \in k$ and $f$ a separable polynomial of degree $3$.

The projective surface $X$, given as the minimal smooth compactification of the affine surface $$X: \quad x^2 - ay^2 = f(z)$$ you have written down is an example of a Châtelet surface. (Note that $X(k) \neq \emptyset$ always as there is a rational point at infinity). 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.

Such surfaces are non-rational provided $a$ is not a square in any of the residue fields of the irreducible factors of $f$. Moreover, in the above paper, it is shown that they are stably rational provided certain assumption hold (e.g. $f$ is irreducible with Galois group $S_3$). But as remarked in the comments, there are examples which are also not stably rational.

They prove this using universal torsors $$T \to X.$$ For a good overview of the theory of universal torsors, I would recommend the book

Skorogobatov - Torsors and rational points

I will just remark that these are torsors under the Néron-Severi torus, in particular the generic fibre is geometrically integral.

A sufficient criterion for the existence of a universal torsor is $X(k) \neq \emptyset$; but as already explained we have this property so universal torsors exist. There may be many universal torsors in general; but the twists of a given torsor gives a parametristation of the rational points of $X$. So there is always some torsor with a rational point. But it turns out that such torsors $T$ are birational to a complete intersection of two quadrics in projective space, which is shown to be a rational variety (details in the above paper). So this shows that (3) holds.

Altogether, this shows that there are Châtelet surfaces which satisfy (3), (5), but not (6), and also those which satisfy (3) but not (5) nor (6). This seems to give a complete answer to your question.

Further details on these results and constructions can be found in the seminar Bourbaki report:

Laurent Moret-Bailly - Variétés stablement rationnelles non rationnelles