  We make use of generalized iterations of a version of the Jensen forcing to
define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of
the following four Choice hypotheses:
  (1) $\text{DC}(\mathbf\Pi^1_n)\land\neg\text{AC}_\omega(\varPi^1_{n+1})\,;$
  (2) $\text{AC}_\omega(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land
\neg\text{AC}_\omega(\mathbf\Pi^1_{n+1});$
  (3)
$\text{AC}_\omega\land\text{DC}(\mathbf\Pi^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$
  (4)
$\text{AC}_\omega\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbf\Pi^1_{n+1}).$
  Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions
(1)--(4) is consistent with ZF.
  As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic
without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a
cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal
P(\omega)$ in this model, such that $\langle\omega, M\rangle$ satisfies
  (5) PA$^0_2$ + $\text{AC}_\omega(\varSigma^1_{\infty})$ +
$\text{CA}(\mathbf\Sigma^1_{n+1})$ + $\neg\text{CA}(\mathbf\Sigma^1_{n+1})$.
  Thus $\text{CA}(\mathbf\Sigma^1_{n+1})$ does not imply
$\text{CA}(\mathbf\Sigma^1_{n+2})$ in PA$^0_2$ even in the presence of the full
parameter-free (countable) Choice $\text{AC}_\omega(\varSigma^1_{\infty}).$
