This probably falls short of an answer, more like a rambling sequence of comments.

Regarding question 2, notice that $E^r$ is an additive functor. I.e., $E^r(X\vee Y)\cong E^r(X)\oplus E^r(Y)$. This implies that if $E^r(X)$ is realized as the homotopy groups of an $n$-excisive functor for some $n$, then it would be a linear functor, i.e, $n$ would be 1. But in this case $E^r(X)$ would be a generalized homology theory, which (unless I am very confused) it is not for $r>2$. So the answer to question 2 is No.

The moral that I would draw from this is that while there may be interesting connections between Goodwillie towers of specific functors and higher cohomology operations, a very general connection between $r$-excisive functors and higher cohomology operations or higher pages in a spectral sequence is perhaps too much to expect. In particular, the Atiyah-Hirzebruch spectral sequence does not appear to have much to do with higher excision.

**Added later**: A particular example of a Goodwillie tower that obviously has a lot to do with homology operations is the tower of the functor from spectra to spectra $X\mapsto H \wedge \Omega^\infty X$, where $H$ is the spectrum representing your favorite homology theory. The tower of this functor calculates the homology of $\Omega^\infty X$ in terms of the homology of $X$. The differentials have a lot to do with homology operations, specifically Dyer-Lashof operations. All this was studied in considerable detail (for $H=H\mathbb Z/2$) in the following paper of N. Kuhn and J. McCarty

*The mod 2 homology of infinite loopspaces*,
Algebr. Geom. Topol. 13 (2013)

Regarding question 1, you could say that the answer is yes, for trivial reasons, if you interpret broadly enough what it means to be "defined using primary cohomology operations". A $2$-excisive functor is the homotopy fiber of a map between two spectra, so if any map between spectra is a cohomology operation, then a functor is determined by a cohomology operation.

To be more specific and a little less trivial, let me recall a general description of $2$-excisive functors. We restrict ourselves to pointed functors between the categories of pointed spaces or spectra. Then any quadratic functor is the homotopy fiber of a natural transformation of the following form, where $C$ is a spectrum and $D$ is a spectrum with an action of $\Sigma_2$:
$$
C\wedge X\to (D\wedge X^{\wedge 2})_{h\Sigma_2}
$$
(in the case of space-valued functors you should put $\Omega^\infty$ in front of these functors, and there may be non-deloopable natural transformation. These correspond to the unstable homotopy operations that were mentioned in a comment).

Natural transformations of this form are well-understood, and can be described in terms of maps from $C$ to (some spectrum constructed out of) $D$. If you call these maps cohomology operations, then you can say that $2$-excisive functors with prescribed first and second layers are classified by cohomology operations of certain type from the first layer to the second.

In case of functors from/to the category of spaces, the derivatives have the structure of a module over the spectral Lie operad. There is a relationship between this module structure and the differentials in the Goodwillie spectral sequence. So you can say that to some extent the cohomology operations that determine the differentials in the Goodwillie spectral sequence come from this Lie module structure. But this is not the whole story.

unstable operations for stable co/homology theories? (This is just reckless speculation). $\endgroup$