I will work stably: everything in sight will be a spectrum.

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It is well known and classical that cohomology operations correspond to map of spectra: that is if $E,F$ are two spectra, a natural transformation $E^*X→F^{*+n}X$ correspond by Yoneda to a map of spectra $E→\Sigma^nF$.

On the other hand if you have a map of spectra $E→\Sigma^nF$, you also get a homology operation $E_*X→F_{*-n}X$. This is because homology can be written as
$$ E_*X = \pi_*(E\wedge X),\qquad F_{*-n}X=\pi_*(\Sigma^nF\wedge X)\,.$$
So understanding cohomology operations will give you also a supply of homology operations. I ignore if those are all of them (and if I had to bet I'd say no, but I cannot substantiate these feelings). This is **not** a dualization as you said, it is just a way to get a homology operation out of a map of spectra: no duality required. Moreover we can prove that all homology operations are of this form. In fact any such operation will induce a natural transformation
$$E^{-*}X=E_*DX→F_{*+n}DX=F^{-*-n}X$$
when $X$ varies among finite spectra and $D$ is the Spanier-Whitehead dual, and we know that these operations are induced by maps $E→\Sigma^nF$. Basically, when you restrict to finite spectra homology and cohomology are pretty much the same thing: you just need a Spanier-Whitehead dual to pass from one to the other.

Thankfully the second differential in the AHSS is always a homology operation of this kind. The quickest way to see this is by looking at the construction of the AHSS that uses the Postnikov tower for the spectrum $E$ instead than the one using the cellular filtration for $X$ (this will also show that the differential $d_2$ correspond essentially to the k-invariants of $E$). That is, we can look at the AHSS as the spectral sequence associated to the exact couple

$$\require{AMScd}
\begin{CD}
\dots @>>> \pi_*(P_{n+1}E\wedge X) @>>> \pi_*(P_nE\wedge X) @>>> \pi_*(P_{n-1}E\wedge X) @>>> \cdots\\
{} @VVV {} @VVV {} @VVV {}\\
\dots @. \pi_*(H\pi_{n+1}E \wedge X)=H_*(X;\pi_{n+1}E) @. \pi_*(H\pi_nE\wedge X)=H_*(X;\pi_nE) @. \pi_*(H\pi_{n-1}E\wedge X)=H_*(X;\pi_{n-1}E) @.\cdots
\end{CD}$$

That is the $d_2:H_*(X;\pi_nE)\to H_{*-1}(X;\pi_{n+1}E)$ is induced by the map of spectra
$$H\pi_nE\to \Sigma P_{n+1}E\to \Sigma H\pi_{n+1}E\,.$$
So we can leverage our understanding of the cohomology operations to get at the $d_2$ of the AHSS.


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Now let us tackle homology cooperations. These are the homotopy classes of $E\wedge E$. I really do not have too much to say about them: they show up because they are important for the Adams spectral sequence (in fact they basically run the show there). They are also the "dual" of the cohomology operations. If $E$ is an $E_\infty$-ring spectrum we can use the universal coefficient spectral sequence to go
$$\mathrm{Ext}_{E_*}(\pi_*(E\wedge E),E_*) \Rightarrow \pi_*F_E(E\wedge E,E)=\pi_*F(E,E)$$
where $F_E$ and $F$ are the internal hom in $E$-modules and spectra respectively. For example when $E=Hk$ for $k$ a field (the main case people treat I think) this spectral sequence degenerates telling you that the $k$-linear dual of $\pi_*(Hk\wedge Hk)$ is the ring of cohomology operations. As you can see these have no direct relation to homology operations and they arise in a different setting altogether.


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Finally, the Hurewicz homomorphism. This is, by definition, the map on homology theories defined by the map $h:\mathbb{S}\to H\mathbb{Z}$ sending 1 to 1. In fact, if $f:\Sigma^n\mathbb{S}\to X$ is a class in $\pi_nX$ we are sending it to the image of the fundamental homology class of $\Sigma^n\mathbb{S}$. But the fundamental homology class is just the map
$$\Sigma^nh=(h\wedge 1):\Sigma^n\mathbb{S}\to H\mathbb{Z}\wedge\Sigma^n\mathbb{S}\,.$$
So we are sending $f$ to $(1_{H\mathbb{Z}}\wedge f)\circ (h \wedge 1) = (h\wedge f)$, that is we are doing the homology operation represented by $h$.