@Burak: nice, I haven't heard of that selection theorem before.

@JoelDavidHamkins: thanks, fixed that.

@JoelDavidHamkins: indeed, that's what I mean. I've updated this phrase again to avoid confusion.

@JoelDavidHamkins: hopefully I've made it less confusing. Statement regarding sufficiency is about sufficient conditions for the existence of a Borel selector. For equivalence of surjective reduction and existence of Borel selectors: the latter implies existence of a Borel transversal which we can take as $Z$. I think the former implies the latter, but I'm still working the proof - so "apparently" so far. Any doubts regarding that statement?

I've also found the flaw in my quotient space argument, summarized it all in the update to the question. Your answer is now complete for me, so thanks a lot again.

In Kechris, a selector of $\sim$ is $h:X\to X$ such that $h(x)\sim x$ and such that $x\sim x'$ implies $h(x) = h(x')$. I guess, the existence of a Borel selector is exactly the existence of a surjective Borel reduction. There are some sufficient conditions in the book, and it may be easier to look for other using this terminology.

The quotient space is not Borel, but I thought it is analytic, then we could take $Z$ being any Borel superset of $X/\!_\sim$. Anyways, let me think of your formulation regarding surjectivity - I'll also check what's in Kechris on Borel ERs, maybe 18.20 is of help here. Can you update the link to notes of Thomas and Schneider, please? Currently it opens Hjorth's notes. I've edited the link, so just wanted to be sure that's the one you've meant.

Thank you, could you suggest then what is incorrect in my construction of $Z$ via a one-point compactification of $X/\!_\sim$ endowed with a quotient topology? Also, do I understand this correctly - if $\sim$ is smooth, still there may not be a surjective version of $g$?

I think this question of mine is related, though I am not sure whether the maximal/minimal $\sigma$-algebra would work there.