What follows is a long string of comments I hope might help to put your questions in context.
All the spaces considered are Polish (separable and completely metrizable). The derived Borel spaces $(X,B(X))$ are called standard Borel spaces. One of the main and basic theorems is that any two uncountable, standard Borel spaces are isomorphic. Going back to the topological spaces, any for any two uncountable Polish spaces $X,Y$ there exists a Borel map $f:X\to Y$ with a Borel inverse.
That actually show that the Borel sets of any two of the aforementioned spaces behave the same. With a little more care, you can calculate the complexity of each of those isomorphisms and, for instance, if the preimage of an open subset of $Y$ by $f$ is a $\Sigma^0_\xi$ subset of $X$, then the Borel sets of $Y$ are brought back to Borel sets of $X$ of at most $\xi$ levels up in complexity (namely, if $f : {X} \rightarrow {Y}$ is $\Sigma_{\xi}^{0}$-measurable and $P$ is a $\Sigma_{\eta}^{0}$ then $f^{-1}[P]$ is $\Sigma_{\xi+\eta}^{0}$; see [Moschovakis, 1G.7]). Moreover, each level of the projective hierarchy is preserved by such an isomorphism [ibid., 1G.1].
Concerning measure, note that any two measure spaces of the form $(X,B(X),\mu)$ where $(X,B(X))$ is standard Borel and $\mu$ is a probability measure, are isomorphic. And since Lebesgue measure on $\mathbb{R}$ is $\sigma$-finite, it is equivalent to a probability measure (i.e. has the same null sets). Therefore all the questions of $\mu$-measurability have the same answer across the whole range of Polish spaces. Incidentally, as in your argument of the similarity between the Baire space and the reales, $^\omega\omega$ is also homeomorphic to a co-countable subset of the Cantor space: Just take the set of sequences of with infinitely many ones.
Concerning RH, one form of the statement depends on the topology, but it can be proved that it is equivalent to another one involving only the counting of primes. And actually, that version is $\Pi^0_1$ (a fortiori, talking only about the first order theory of natural numbers), and hence it is absolute.
Best Answer
I'll answer about the boldface versions.
The ambiguous class $\boldsymbol\Delta_2^0$ comprises $\omega_1$ levels of the Wadge hierarchy (see Kecrhis' book on descriptive set theory, Exercise 21.16, and the many references therein).
Sets in $\boldsymbol\Sigma_\alpha^0 \setminus \boldsymbol\Pi_\alpha^0$ are complete, and from this it follows that they are all inter-reducible, and hence in a unique degree.
Finally, all the results hold uniformly of all zero dimensional Polish spaces, in particular $A^\omega$ for any countable $A$.
You might also be interested in the comment by A. Caicedo to this question.