Wow, that's eight questions, plus more in the comments -- I don't think I can answer all of them, but I'll try to answer at least a few! :)
First of all, let's fix the setting: It seems to me that you are using three different kinds of level structures, $\Gamma_0(p^n)$, $\Gamma_1(p^n)$ and $\Gamma(p^n)$, and some questions seem to be referring to different ones.
Also, there are various possible different meanings of $\mathcal X$ and $\mathfrak X$, since these mean different things in each paper respectively.
So we first have to agree on some uniform setting for your questions: Since you are interest in perfectoid moduli spaces, I suggest we follow Scholze (III.2.2 in the torsion paper) and denote by $\mathfrak X$ the formal completion of the modular curve over $\mathbb Z_p^\mathrm{cyc}$ of some fixed tame level. Denote by $\mathfrak X^{\ast}$ the completion of the compactified modular curve, and by $\mathcal X^{\ast}$ the adic generic fibre. Since you are interested in moduli of elliptic curves, I suggest we now deviate from Scholze's notation and denote by $\mathcal X$ the analytification of the modular curve over $\mathbb Q^{\mathrm{cyc}}$ (rather than the generic fibre of $\mathfrak X$, which is the good reduction locus).
moduli interpretations of the spaces $\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a$, $\mathcal X_{\Gamma_1(p^\infty)}(\epsilon)_a$ and $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a$
(Most of what I'm going to say in regards to this question can be found in more detail in this related article: https://nms.kcl.ac.uk/ben.heuer/PGp-adMC.pdf.)
Let's follow the torsion paper and start with level $\Gamma_0(p^n)$ and the anticanonical locus $\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a$ of some tame level. This represents the functor which sends a (sheafy) adic space $\mathrm{Spa}(R,R^{+})$ to the set of isomorphism classes of triples $(E,\alpha,D)$ where $E|R$ is an elliptic curve with some condition on the Hasse invariant which ensures that $E$ has a canonical subgroup $C=C(E)\subseteq E[p]$, where $\alpha$ is a tame level structure, and where $D\subseteq E[p^n]$ is an anticanonical cyclic subgroup scheme of rank $p^n$. Here "anticanonical" means $D\cap C=0$.
Scholze now proves that there is a perfectoid space $$\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a.$$ Since perfectoid tilde-limits satisfy the universal property of the limit for perfectoid test objects, this space represents the functor which sends $\mathrm{Spa}(R,R^{+})$ for any perfectoid $\mathbb Q_p^\mathrm{cyc}$-algebra $R$ to the set of isomorphism classes of $(E,\alpha,D_\infty)$ where $D_\infty = (D_n\subseteq E[p^n])_{n\in\mathbb N}$ is a collection of anticanonical cyclic subgroup schemes with $D_{n+1}[p^n]=D_n$ (See Corollary 3.2 of the above document). So in regards to your question of moduli of $p$-divisible groups, one could call this data an "anticanonical $p$-divisible subgroup of height 1".
Similar results hold for the perfectoid tilde-limits $\mathcal X_{\Gamma_1(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma_1(p^n)}(\epsilon)_a$ and $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma(p^n)}(\epsilon)_a$ by the same reasoning: The first represents the functor which sends $\mathrm{Spa}(R,R^{+})$ for any perfectoid $\mathbb Q_p^\mathrm{cyc}$-algebra $R$ to the set of isomorphism classes of $(E,\alpha,\beta: \mathbb Z_p\xrightarrow{\sim} T_pD_\infty(R))$ where $D_\infty$ is an anticanonical $p$-divisible subgroup of height 1 and beta is a trivialisation of its Tate module. The second uses instead isomorphism classes of tuples $(E,\alpha,\gamma: \mathbb Z_p^2\xrightarrow{\sim} T_pE(R))$ where the image of $\gamma(1,0)$ in $E[p](R)$ generates an anticanonical subgroup.
the formal model of the anticanonical tower
As with the last question, we first need to agree on a base: The torsion paper considers an anticanonical tower over $\mathbb Z_p^\mathrm{cyc}$ (whose limit you denote by $\mathfrak X_\infty(\epsilon)$), whereas Le halo spectral basically works over $\mathbb Z_p$ (as you say, they really work relatively to some weight space, which is great because it allows them to construct integral families of modular forms. But I think in order to understand what's going on in terms of moduli, it might be easier if we specialise to a point -- the weight space doesn't change much in that respect). Let's follow Scholze and work over $\mathbb Z_p^\mathrm{cyc}$ if you don't mind, so we simply base-change the constructions of Andreatta--Iovita--Pilloni to $\mathbb Z_p^\mathrm{cyc}$ (their constructions actually require Noetherianess in several places in order to construct normalisations, but once you got the spaces, you may still simply base-change to $\mathbb Z_p^\mathrm{cyc}$. The resulting spaces agree with Scholze's $\mathfrak X^{\ast}(\epsilon)$ up to a normalisation issue).
Now there are arguably two "anticanonical towers", which are isomorphic: The first one, which gives the tower its name, is the tower
$$\dots\to\mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a\to\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a\to \mathcal X^{\ast}(\epsilon).$$
The second tower is used in the torsion paper to prove that the above tower has a perfectoid tilde limit $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$: Let's recall how this works. Let $\mathfrak X^{\ast}(\epsilon)$ be like in Scholze's Definition III.2.12. As explained there, (away from the cusps) this represents the functor sending $\mathrm{Spf}(R)$ for $p$-adically complete $\mathbb Z_p^\mathrm{cyc}$-algebras $R$ to the set of isomorphism classes $(E,\alpha,\eta)$ where $E|R$ is an elliptic curve, $\alpha$ is a tame level and $\eta\in \omega_E^{\otimes(1-p)}$ such that $\eta \mathrm{Ha} = p^{\epsilon} \in R/p$.
Scholze constructs Frobenius lifts $F:\mathfrak X^{\ast}(p^{-1}\epsilon)\to \mathfrak X^{\ast}(\epsilon)$ which on the level of moduli (away from the cusps) are given by quotienting by the canonical subgroup, i.e. sending $E\mapsto E/C$.
In the limit, this gives rise to the space $\mathfrak X^{\ast}(p^{-\infty}\epsilon)=\varprojlim_{{F}} \mathfrak X^{\ast}(p^{-n}\epsilon)$ which is integrally perfectoid. In particular, its adic generic fibre is a perfectoid space.
The relation to the anticanonical tower is that on the level of adic spaces over $\mathbb Q_p^\mathrm{cyc}$, there is a natural "Atkin-Lehner" isomorphism
$$\varphi_n:\mathcal X^{\ast}(p^{-n}\epsilon)\to \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a, \quad E\mapsto (E/C_{n},E[p^n]/C_{n})$$
where $C_n\subseteq E[p^n]$ is the rank $p^n$ canonical subgroup. Its inverse is given by sending $(E,D)\mapsto E/D$.
One can now check on the level of moduli that for different $n$, these give a comparison isomorphism between the anticanonical tower and the Frobenius tower:
$\require{AMScd}$
\begin{CD}
\dots @>>> \mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a @>>> \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a @>>> \mathcal X^{\ast}(\epsilon)\\
@AAA @AA\varphi_2A @AA\varphi_1A @|\\
\dots @>>> \mathcal X^{\ast}(p^{-2}\epsilon) @>>> \mathcal X^{\ast}(p^{-1}\epsilon) @>>> \mathcal X^{\ast}(\epsilon)
\end{CD}
We may thus see the tower of morphisms $F:\mathfrak X^{\ast}(p^{-(n+1)}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$ as a formal model of the anticanonical tower. In particular, we may see $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$ as a canonical formal model for $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$. Alternatively, I think this should imply that we can regard $\mathfrak X^{\ast}(p^{-n}\epsilon)$ as representing (away from the cusps) tuples $(E,\alpha,D_n)$ where $D_n\subseteq E[p^n]$ is a cyclic rank $p^n$ subgroup scheme which is generically anticanonical (its special fibre may well be canonical).
So what is the moduli interpretation of a $\mathrm{Spf}(R)$-point of $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$ (away from the cusps) where $R$ is a complete $\mathbb Z_p^\mathrm{cyc}$-algebra? One answer is that, by definition, it is the data of $(E_0,E_1,E_2,\dots,\alpha, (\eta_n)_{n\in\mathbb N})$ where $(E_0,\alpha,\eta_0)$ is like before, $E_{n+1}/C(E_{n+1})=E_n$ for all $n$, and the $\eta_n$ are compatible under $F$. Alternatively, by the above tower this should be equivalent to the data of $(E,\alpha, (\eta_n)_{n\in\mathbb N},D_\infty)$ where $E:=E_0$ and $D_\infty=(D_n)_{n\in\mathbb N}$ is a generically anticanonical $p$-divisible subgroup of $E[p^\infty]$ of height 1. Here $D_n$ is defined as the kernel of the dual isogeny to $E_n\to E_0$, so that $E_n=E_0/D_n$, and the $\eta_n\in \omega_{E_n}^{\otimes(1-p)}$ are as before.
the integral model for $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ of Andreatta--Iovita--Pilloni
Now to the spaces in Le halo spectral, I'll try to elaborate on Leeeeroy_Jennnnkins' answer and answer a question raised in the comments. (If you allow another plug, most of this can be found in more detail in \S 4 of https://arxiv.org/pdf/1902.03985.pdf).
Andreatta--Iovita--Pilloni go further than what you denote by "$\mathfrak X_\infty(\epsilon)$": They also consider the Igusa schemes $\mathfrak I\mathfrak G_n(p^{n}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$ which relatively represent the choice of a trivialisation $\mathbb Z/p^n\mathbb Z\to C_n^\vee$, namely morphisms which are an isomorphism over the ordinary locus. They show that the Frobenius isogeny lifts to a "Frobenius" morphism $F:\mathfrak I\mathfrak G_{n+1}\to \mathfrak I\mathfrak G_n$ and form the "Igusa curve at infinite level" which in order to be consistent with my notation I should probably denote by $\mathfrak I\mathfrak G_{\infty}(p^{-\infty}\epsilon)=\varprojlim_{F}\mathfrak I\mathfrak G_{n}(p^{-n}\epsilon)$.
Now how does this compare to Scholze's spaces? The short exact sequence of group schemes
$$0\to C_n\to E[p^n]\to E[p^n]/C_n\to 0$$
shows that the Weil pairing canonically identifies $C_n^{\vee}$ with $E[p^n]/C_n$. Thus the Igusa tower equivalently parametrises trivialisations $\mathbb Z/p^n\mathbb Z\to E[p^n]/C_n$. But under the above "Atkin-Lehner" isomorphism, $E[p^n]/C_n$ is the corresponding anticanonical subgroup of $E/C_n$. This means that on the adic generic fibre, this isomorphism lifts to a canonical isomorphism
\begin{CD}
\mathfrak I\mathfrak G_n(p^{-n}\epsilon)^{\mathrm{ad}}_{\eta} @>\sim>>\mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \\
@AAA @AAA \\
\mathcal X^{\ast}(p^{-n}\epsilon) @>\sim>> \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a.
\end{CD}
In particular, this means that $\mathfrak I\mathfrak G_n(p^{-n}\epsilon)$ is a formal model of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$. In the limit, it follows that
$$\mathfrak I\mathfrak G_\infty(p^{-\infty}\epsilon)^{\mathrm{ad}}_{\eta}=\mathcal X^{\ast}_{\Gamma_1(p^\infty}(\epsilon)_a.$$
So this gives you a canonical formal model of $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$. Its moduli interpretation (away from the cusps) may be given in terms of tuples $(E,\alpha,(\eta_n)_{n\in\mathbb N},\beta:\mathbb Z_p\to T_pD_\infty)$ where $(E,\alpha,(\eta_n)_{n\in\mathbb N},D_\infty)$ is like above, and beta is a morphism that becomes an isomorphism over the ordinary locus.
Finally, if you are interested in integral models for the full level modular curve $\mathcal X^{\ast}_{\Gamma(p^\infty)}$, you may want to have a look at Lurie's preprint http://www.math.harvard.edu/~lurie/papers/LevelStructures1.pdf.
universal elliptic curves
There are different universal elliptic curves over $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$, and the "right" one depends on your choice of moduli interpretation. Looking at the above comparison map of towers again, as you say, we get a different "universal elliptic curve" by pullback along any $\mathfrak X^{\ast}(p^{-\infty}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$. This is the universal $E_n$ in the moduli description in terms of data $(\alpha,\eta,E_0,E_1,E_2,\dots)$. Alternatively, the moduli interpretation in terms of $(E,\alpha,\eta,D_\infty)$ suggest to look at the pullback $\mathfrak E_\infty^{\mathrm{univ}}$ of the universal elliptic curve $\mathfrak E^{\mathrm{univ}}$ along $\mathfrak X^{\ast}(p^{-\infty}\epsilon)\to \mathfrak X^{\ast}$.
Can we make sense of the adic generic fibre of $\mathfrak E_\infty^{\mathrm{univ}}\to \mathfrak X^{\ast}(p^{-\infty}\epsilon)$? Yes:
The adic generic fibre of $\mathfrak E_\infty^{\mathrm{univ}}$ can be described as the fibre product of the relatively smooth rigid space $(\mathfrak E^{\mathrm{univ}})^{\mathrm{ad}}_{\eta}\to \mathcal X^{\ast}(\epsilon)$ with the perfectoid space $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)\to \mathcal X^{\ast}(\epsilon)$. I think this should exist as a sousperfectoid (hence sheafy) adic space $\mathcal E_\infty^{\mathrm{univ}}$.
Is it a perfectoid space over $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$? No:
Fibre products of perfectoid spaces are perfectoid, but if you take the fibre product with any point $\mathrm{Spa}(\mathbb Q_p^\mathrm{cyc})\to \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)$, you will get the analytification of an elliptic curve $E^{an}\to \mathrm{Spa}(\mathbb Q_p^{\mathrm{cyc}})$ which is certainly not perfectoid. If you want something perfectoid, I think it would be reasonable to guess that $\varprojlim_{[p]} \mathcal E_\infty^{\mathrm{univ}}$ is perfectoid -- this is true over the good reduction locus, but as far as I know, it's not currently known whether it's true over the whole space.
