I think there is some vagueness inherent in the "similarly define ...". How is one to assign the consistency statement $Con(ZF_\lambda)$ for computable $\lambda$? This looks trivial but it is not.
I think also ZF is something of a red herring here. The question arises in PA (since we are looking at $\Pi_1$ sentences quantifying over natural numbers.)
Feferman has shown ("Transfinite Recursive Progression of Theories" JSL 1962) that it is possible to assign for every n in an effective manner a $\Sigma_1$-formula $\varphi_n(v_0)$ where each of the latter is to be thought of as enumerating (integer codes of) axiom sets (which I'll call "theories."). This is done in such a fashion so that if $a,b$ are integers with $b = 2^a$ that $T_b$ is $T_a$ together with the statement
$$\forall \psi \in \Sigma_1\forall x [ Prov_{T_a}\psi(x) \longrightarrow \psi(x)]$$
(This is thus a "1-Reflection Principle" - for $\psi\in\Sigma_1$ here). He does this
with a view to considering those integers $a$ that are notations for recursive ordinals (in the sense of the notation system devised by Kleene - "Kleene's $O$".)
(There are clauses for $a$ representing a notation for a limit ordinal, when $a = 3^e$).
He proves that there are linear paths through the system of notations of computable ordinals, going through all recursive ordinals $\alpha$,so that
Every true $\Pi_2$
sentence in arithmetic is proven by one of the theories along the path.
The starting
theory $T_0$ here can be PA (or ZFC if you want). Such a path gives
a definite meaning to $ZF_0, \ldots, ZF_\alpha, \ldots$ etc. for recursive $\alpha$.
Moreover for such a particular progression of theories one would would construe the answer to the question to be "No".
Feferman's starting point was the 1939 paper of Turing ("On Systems of Logic Based on Ordinal"). Turing also considered such paths through Kleene's $O$, but could just prove a theorem for $\Pi_1$ sentences, (using simpler "Consistency" statements"). Feferman shows that if one takes "$n$-Reflection" statements
for every $n$ each time one extends the theory then there are paths along which every true statement of
arithmetic is proven.
The moral of the story is that there are very complex ways of simply defining sequences
of theories, (because there are infinitely many ways, or Turing programs, of representing a recursive ordinal) which can hide/disguise all sorts of information.
A very readable survey is Franzen: "On Transfinite Progressions" BSL 2004.
Update (This is an answer to Scott Aaronson's Update.)
He asks:
given a positive integer k, can we say something concrete about which iterated consistency statements suffice to prove the halting or non-halting of every k-state Turing machine?
Let $M_0, \ldots ,M_{n-1}$ enumerate the $k$-state TM's. Let $P$ be the subset of $n$
of those indices of TM's in the list that halt.
The statement
$\forall i (i \in P \rightarrow M_i$ halts $ \wedge \,
i \notin P \rightarrow M_i $ does not halt $)$
is a $\Pi_2$ statement. In Feferman's paper (op.cit.) he shows that every true
$\Pi_2$ statement is proven by a theory $T_a$ in a 1-Reflection sequence, where $a$ is a notation for an ordinal of rank equal to $\omega^2 + \omega + 1 $.
So in terms of the question we do not need to vary the $\alpha$ depending on what
ordinals a $k$-state machine can produce. (Just fix $\alpha$ as given
above.) Of course it gives us zero practical information: there are infinitely many such notations of that rank, and we may not know which one to look at.
