So one of my favorite rant on this topic is that the module point of view is often better suited for these kind of questions.
The module Deuring correspondance is that if $E_0/\mathbb{F}_{p^2}$ is a (maximal) supersingular curve and $O_0=\mathrm{End}(E_0)$, then $E \mapsto M_E=Hom(E, E_0)$ is an anti-equivalence of category between (maximal) supersingular elliptic curves and $O_0$-left torsion free modules of rank $1$ (with the obvious morphisms). As a special case, $E_0$ corresponds to $O_0$. [As an aside, this extends naturally to higher dimension by allowing torsion free modules of rank $g$, while of course the ideal point of view can only describe elliptic curves.]
From this point of view, the natural isomorphism $End(E) = End(M_E)$ is given as follows: if $\gamma \in End(E)$ and $\phi: E \to E_0$, then $\phi \cdot \gamma$ is the composition $\phi \circ \gamma$. (Note that $End(E)$ acts on the right of $M$, and $M$ has a natural $(O_0, End(E))$-bimodule structure: if $\gamma_0 \in O_0$, $\gamma_0 \cdot \phi = \phi \circ \gamma_0$.) This answers the first question.
For the second question: if $\phi: E_1 \to E_2$ is an isogeny, then the associated module monomorphism $M_{\phi}: M_2 \to M_1$ is given in the same manner as the construction above: to a morphism $\psi: E_2 \to E_0$ we associate $\psi \circ \phi: E_1 \to E_0$. In particular, the image of $M_2$ in $M_1$ are given by all morphisms $E_1 \to E_0$ that factor through $\phi$.
As a special case, an isogeny $\phi: E_0 \to E$ corresponds to a module monomorphism $M_E \to O_0$, and the image of $M_E$ is the ideal $I_E$ associated to $E$ by the "ideal Deuring correspondance". We recover Luca's point of view that $I_E$ is the set of all endomorphisms of $E_0$ that factor through $\phi$. But note that $I_E$ is not intrinsic (it depends on the choice of $\phi$), while $M_E$ is intrinsic. That's why it is often easier to reason on $M_E$ directly.
So now let's apply this to $\phi_{chl} \circ \phi_{sk}$: we have an associated module morphism $M_{chl} \to O_0$, whose image is, by the discussion above, precisely the ideal of endomorphisms of $E_0$ that factor (at the start) through $\phi_{chl} \circ \phi_{sk}$. Likewise for $I_{com}$, and so its conjugate is the ideal of endomorphism that factor at the end through $\hat{\phi}_{com}$. Assuming that the degree of these isogenies are coprime, it means that $\gamma \in O_0$ is in the intersection if it can be written as $\gamma = \hat{\phi}_{com} \circ \gamma' \circ \phi_{chl} \circ \phi_{sk}$ for some $\gamma': E_{chl} \to E_{com}$.
From the module point of view, the intersection $\overline{I}_{com} \cap I_{sk} \cdot I_{chl}$ is the image of $Hom(M_{com}, M_{chl})$ given by $(\psi: M_{com} \to M_{chl}) \mapsto M_{\phi_{chl} \circ \phi_{sk}} \circ \psi \circ M_{\hat{\phi}_{com}}$ and so the endomorphism $\gamma \in O_0$ is in this intersection if it factors as $\gamma: O_0 \to M_{com} \to M_{chl} \to M_{sk} \to O_0$. (This is just a restatement of the discussion above on isogenies, using contravariance). But such a $\gamma$ also gives an endomorphism of $M_{com}$ (by shifting the individual isogenies), that factor through $\phi_{chl}\circ \phi_{sk} \circ \hat{\phi}_{com}$.
