There are two issues. One I understand and one I don't. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.

(EDIT:) $T$ will correspond to the lattice $\Lambda$ in $\mathbb{C}$ generated by all modular symbols. This is because $V_{2,\mathbb{Z}}(f)$ is the image of $V_{2,\mathbb{Z}} (Y_1(N)) = H_1 \bigl( X_1(N)(\mathbb{C}),\{\text{cusps}\},\mathbb{Z}\bigr)$ inside $H_1(E(\mathbb{C}),\mathbb{Z})$. The lattice $\Lambda$ will contain the lattice $\Lambda_0$ of the $X_0$-optimal (strong Weil) curve $E_0$, but it may be strictly larger if $E_0$ has rational torsion points, e.g. for 11a. Note also that $\Lambda$ need not be the lattice of an elliptic curve in the isogeny class, but rather $\tfrac{1}{2}\Lambda_E$ for some elliptic curve $E$. in case $E_0[2] \subset E_0(\mathbb{Q})$, e.g. for 17a. But can only happen for $2$-torsion. So if $p>2$, then $T$ is the Tate-module of an elliptic curve $E_*$ which is an étale quotient of $E_0$. The kernel of $E_0 \to E_*$ will be trivial or $\mathbb{Z}/p\mathbb{Z}$. So let $E$ be this curve for the rest of the answer.

There are two kinds of them. The $z_{\gamma}$ and the ${c,d}$ $z_{m}$. (sorry I don't seem to be able to produce indices before the symbol inMathJax) The latter are in $H$, see 8.1 of Kato, but they depend on the choices of $c$ and $d$. They are useful for bounding the Selmer group as, for a fixed $c$ and $d$ they form an Euler system.

The $z_{\gamma}$ instead is linked to the $p$-adic $L$-function and they are independent of the choices. They are obtained by dividing by $\mu(c,d)$, page 229 of Kato. So they need not be integral anymore. The appendix A in Delbourgo's book "Elliptic curves and big Galois representations" discusses this in detail. The fact that $z_\gamma$ is not known to be integral is linked to the integrality of the $p$-adic $L$-function (the vanishing of the $\mu$-invariant is an even harder question, I believe). But even the integrality of the $p$-adic $L$-function (known for the $X_1$-optimal curve by Greenberg-Vatsal) does not seem to imply that $z_\gamma$ is integral in $H$, maybe it does in $H^1_{\mathrm{Iw}}(\mathbb{Q}_p,T)$.

To prove that $z_{\gamma}$ is integral, one uses that $H$ is a free $\Lambda$-module of rank 1, e.g. as shown in 12.4.(3) if $T/pT$ is irreducible. In fact it is not hard to show that $H$ is free also if $E(\mathbb{Q})[p]$ is trivial. For instance, we really only have to worry when $T/pT$ is irreducible rather than just whether the representation is surjective.

If $T$ were known to be $T_pE$ as in (1), then my corrections aimed to show that one can still deduce the divisibility, because the denominators introduced by $\mu(c,d)$ are not too big, i.e. $Z(f,T)$ has finite index in $Z$, see 13.12 in Kato. This would also prove that the $p$-adic $L$-function is integral and that $\mu\geq 0$, but it would not say anything about $\mu=0$.

Now, in the proof of the divisibility in the main conjecture as in 12.5.(3), one also needs to apply the Euler system method. And here we need the element $\binom{1\ 1}{0\ 1}$ in the image of the Galois representation, see Hyp($K_{\infty},T)$ in Rubin's book. So the integrality won't help, yet. My flawed paper was based on the fact that if $E[p]$ is reducible, then we can circumvent the problem in the Euler system argument, by knowing that the $\mu$-invariant for class groups vanish.

In summary, I am certain everything is integral if $T/pT$ is irreducible and I could imagine that one can prove it too in the reducible case, maybe assuming Steven's conjecture on $\mu=0$. For Kato's divisibility on the other hand, the surjectivity of the representation to $\mathrm{GL}_(\mathbb{Z}_p)$ is still needed.

There are two issues. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.

$T$ will correspond to the lattice $\Lambda$ in $\mathbb{C}$ generated by all modular symbols. This is because $V_{2,\mathbb{Z}}(f)$ is the image of $V_{2,\mathbb{Z}} (Y_1(N)) = H_1 \bigl( X_1(N)(\mathbb{C}),\{\text{cusps}\},\mathbb{Z}\bigr)$ inside $H_1(E(\mathbb{C}),\mathbb{Z})$. The lattice $\Lambda$ will contain the lattice $\Lambda_0$ of the $X_0$-optimal (strong Weil) curve $E_0$, but it may be strictly larger if $E_0$ has rational torsion points, e.g. for 11a. Note also that $\Lambda$ need not be the lattice of an elliptic curve in the isogeny class, but rather $\tfrac{1}{2}\Lambda_E$ for some elliptic curve $E$. in case $E_0[2] \subset E_0(\mathbb{Q})$, e.g. for 17a. But can only happen for $p=2$ and for $p$ of additive reduction. So if $p$ is a prime of odd semi-stable reduction, then $T$ is the Tate-module of an elliptic curve $E_*$ which is an étale quotient of $E_0$. So let $E$ be this curve for the rest of the answer.

There are two kinds of them. The $z_{\gamma}$ and the ${c,d}$ $z_{m}$. (sorry I don't seem to be able to produce indices before the symbol in MathJax) The latter are in $H$, see 8.1 of Kato, but they depend on the choices of $c$ and $d$. They are useful for bounding the Selmer group as, for a fixed $c$ and $d$ they form an Euler system.

The $z_{\gamma}$ instead is linked to the $p$-adic $L$-function and they are independent of the choices. They are obtained by dividing by $\mu(c,d)$, page 229 of Kato. So they need not be integral anymore. The appendix A in Delbourgo's book "Elliptic curves and big Galois representations" discusses this in detail. Kato shows that they are in $H\otimes \mathbb{Q}_p$

Kato shows that $z_{\gamma}$ is integral if $H$ is a free $\Lambda$-module of rank 1, e.g. as shown in 12.4.(3) if $T/pT$ is irreducible. In fact it is not hard to show that $H$ is free also if $E(\mathbb{Q})[p]$ is trivial.

Now if the curve admits an isogeny of degree $p$, one can show that for all curves $A$ in the isogeny class $H^1_{\text{Iw}}(\mathbb{Q},T_p A)$ is a free $\Lambda$-module of rank $1$, except for at most a single one of them (I mean up to non-$p$-isogenies of course). This exception - if present - will always be the minimal curve $E_{\text{min}}$ in the class. Moreover if $E_{\text{min}}$ is does not have a free $H^1_{\text{Iw}}$, then there is an embedding of it into $\Lambda$ with image equal to the maximal ideal of $\Lambda$. One can now conclude from the interpolation property of the $p$-adic $L$-function that even if $E_*=E_{\text{min}}$ then $z_{\gamma}$ will be in $H$.

Using this one can prove that the $p$-adic $L$-function is integral and that $\mu\geq 0$, but it would not say anything about Greenberg's $\mu=0$ conjecture. Furthermore one gets a proof of the divisibility in the main conjecture as in 12.5.(3), if the $T/pT$ is reducible. However this conclusion can not be extended at present to all odd semi-stable primes, because there may be primes for which the Galois representation is not surjective, yet $T/pT$ is irreducible; because the Euler system method requires and element $\binom{1\ 1}{0\ 1}$ in the image of the Galois representation, see Hyp($K_{\infty},T)$ in Rubin's book. So the integrality won't help, yet.

In summary, all zeta-elements of Kato are integral with respect to $T$ in the case of an elliptic curve. For Kato's divisibility on the other hand, the surjectivity of the representation to $\mathrm{GL}_(\mathbb{Z}_p)$ or its reducibility is still needed.

(edits: quite a few in the whole answer above, now that I konw the full answer to the question)

There are two issues. One I understand and one I don't. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)$ and $f$ is the modular form associated to the isogeny class of $E$.

I thought $T$ is the Tate-module of the $X_1$-optimal curve, but I was told this is wrong. I truely hope someone will answer this sub-question. Maybe Tony Scholl could say something about it.

There are two issues. One I understand and one I don't. Let $H=H^1_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V_{\mathbb{Z}_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.

(EDIT:) $T$ will correspond to the lattice $\Lambda$ in $\mathbb{C}$ generated by all modular symbols. This is because $V_{2,\mathbb{Z}}(f)$ is the image of $V_{2,\mathbb{Z}} (Y_1(N)) = H_1 \bigl( X_1(N)(\mathbb{C}),\{\text{cusps}\},\mathbb{Z}\bigr)$ inside $H_1(E(\mathbb{C}),\mathbb{Z})$. The lattice $\Lambda$ will contain the lattice $\Lambda_0$ of the $X_0$-optimal (strong Weil) curve $E_0$, but it may be strictly larger if $E_0$ has rational torsion points, e.g. for 11a. Note also that $\Lambda$ need not be the lattice of an elliptic curve in the isogeny class, but rather $\tfrac{1}{2}\Lambda_E$ for some elliptic curve $E$. in case $E_0[2] \subset E_0(\mathbb{Q})$, e.g. for 17a. But can only happen for $2$-torsion. So if $p>2$, then $T$ is the Tate-module of an elliptic curve $E_*$ which is an étale quotient of $E_0$. The kernel of $E_0 \to E_*$ will be trivial or $\mathbb{Z}/p\mathbb{Z}$. So let $E$ be this curve for the rest of the answer.