  Given a function field $K$ and $\phi \in K[x]$, we study two finiteness
questions related to iteration of $\phi$: whether all but finitely many terms
of an orbit of $\phi$ must possess a primitive prime divisor, and whether the
Galois groups of iterates of $\phi$ must have finite index in their natural
overgroup $\text{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated
preimages of $0$ under $\phi$. We focus particularly on the case where $K$ has
characteristic $p$, where far less is known. We resolve the first question in
the affirmative under relatively weak hypotheses; interestingly, the main step
in our proof is to rule out "Riccati differential equations" in backwards
orbits. We then apply our result on primitive prime divisors and adapt a method
of Looper to produce a family of polynomials for which the second question has
an affirmative answer; these are the first non-isotrivial examples of such
polynomials. We also prove that almost all quadratic polynomials over
$\mathbb{Q}(t)$ have iterates whose Galois group is all of $\text{Aut}(T_d)$.
