Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form

Getting rid of the power series, your question boils down to showing that any polynomial $P(x)$ divides some polynomial $Q(x)$ which has the form $Q(x) = \sum_{i} c_i x^{2^i}$. Polynomials $Q$ which have this property are known as additive polynomials. They have some nice alternative descriptions: for instance, if $K$ is a field of characteristic $2$ then a polynomial $Q(x) \in K(x)$ with distinct roots is additive if and only if the set of roots of $Q(x)$ in $\overline{K}$ form a $\mathbb{F_2}$-vector space. (This is a nice exercise, or see Basic Structures of Function Field Arithmetic by Goss for a proof.)

Using this description, it's easy to construct $Q(x)$. Let $K = \mathbb{F_2}(t)$ and let $V$ be the $\mathbb{F_2}$-subspace of $\overline{K}$ generated by the roots of $P$. We set $Q(x) = (\prod_{\alpha \in V} (x-\alpha))^{2^N}$ for $N$ chosen to be large enough so that $P(x)$ divides $Q(x)$. Then $Q(x) \in K(x)$ because the finite vector space $V$ is stable under the action of $\mathrm{Gal}(\overline{K}/K)$, and $Q(x)$ is additive by the fact of the previous paragraph + Frobenius

(Note that everything here generalizes in the obvious way if you replace $\mathbb{F}_2$ by $\mathbb{F}_q$ for any prime power $q$.)

In fact for any integral domain $R$, if $x$ is any element that satisfies a polynomial equation of degree $n$ over $R$, and $S$ is any set of numbers of size greater than $n$, then $x$ satisfies a polynomial equation of the form $\sum_{i \in S} c_i x^i$ for $c_i \in R$ not all zero.

This is just because $\{x^i | i\in S\}$ cannot be linearly independent over the field of fractions of $R$, so some linear relation holds over the field of fractions, and we can clear denominators.