Do these ordinals exist?

For the definition of $F_{n+1}$ to make sense, we need, in addition to the usual axiomatic apparatus of ZFC, a notion of "satisfaction of formulas in $V$." If we have this additional notion and if we allow it to occur in replacement axioms, then we can prove that $F_n(\alpha)$ exists and is countable for every $n$ and $\alpha$, and therefore $F_\omega(\alpha)$ also exists and is countable. The argument is essentially as in Zetapology's answer, with the additional apparatus replacing the "Clearly" claim (which isn't justified without some definability and some use of replacement axioms).

Let $V$ be a transitive model of $ZFC$. Without special assumptions about the model $V$, it is possible that $F_0(0)$ does not exist (in other words - it is possible that every ordinal is definable without parameters). In fact, every model of $ZFC + V=HOD$ has an elementary submodel such that all its elements are definable without parameters - for example, the Skolem closure of the empty set, using the definable Skolem functions. These models are called "Pointwise definable models".

The situation in which $F_0(0)$ exists but $F_n(0)$ does not exist for some $n$ may occur as well. For example, let $V$ be a pointwise definable model of $ZFC + V = HOD$ + there is a measurable cardinal (this is a vast overkill). Let $\kappa$ be a measurable cardinal in $V$ and let $j \colon V \to M$ be the ultrapower embedding by a normal measure on $\kappa$.

Let us claim that $\kappa$ is the first undefinable ordinal in $M$. Indeed, if $\varphi$ was a definition for $\kappa$ in $M$ then, by elementarity, $\varphi$ defines an unique ordinal in $V$, $\gamma$. But this implies that $j(\gamma) = \kappa$ which is impossible.

Every element in $M$ is of the form $j(f)(\kappa)$ for some $f\in V$. Since $f$ is definable without parameters in $V$, $j(f)$ is definable (with the same definition) in $M$. In particular, every element in $M$ is definable from the parameter $\kappa$. Thus, $F_0(0) = \kappa$ and $F_1(0)$ doesn't exist.

One can iterate this process in order to get for every $n < \omega$ a model in which $F_n(0)$ exists while $F_{n+1}(0)$ doesn't exists.