UPDATE: Let me answer the question of where Kato requires the full force of (12.5.2). First of all, there is something tricky in the sense that the answer is "nowhere directly in the article we are discussing". However, this hypothesis is crucial to the method of Euler system, so it is used it in the reference to his prior article Euler systems, Iwasawa theory and Selmer groups (there, it is hypothesis $ii_{str}$).

Finally, though it is true that integrality of Euler systems elements does not in itself imply that $\mu$ vanishes by the usual divisibility, it is nevertheless true that by playing around with reciprocities laws, you can extract the $\mu$-invariant from integrality properties of compatible systems of norms (and in particular Euler systems). This is work of T.Fukaya (unpublished, I think) in the case of $\textrm{GL}_{1}$. Now that the most general form of the reciprocity law is known by the work of Colmez, it might be within reach to link integrality of Kato's Euler system to the $\mu$-invariant of modular forms. Whence my remark to this effect at the end of my original answer. Contrary to the case of the cyclotomic extension, the full-force of this idea can be exploited in the case of the anticyclotomic extension of an imaginary quadratic field (and here again the vanishing of the $\mu$-invariant is almost equivalent to the integrality of the Euler system).

1

First, let us assume that $p$ is odd for safety.

Then I think your impression is correct: the first statement of 12.5 (4), i.e the integrality of the module generated by the $\mathbf{z}^{(p)}_{\gamma}$, is true under the hypothesis that $T/pT$ is irreducible. Hypothesis (12.5.2) is used only to prove the second (and much more spectacular) statement, i.e the Iwasawa-theoretic divisibility.

When $E$ admits a $p$-isogeny, then C.Wuthrich has a result towards integrality. However, the published version is flawed and I admit I can't quite understand the correction on his webpage. Because C.Wuthrich is a frequent poster here, we can count on his explanation.

In general, proving that Euler systems are integral should be quite a difficult task because it is generally intimately linked with the question of whether the $\mu$-invariant vanishes.