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For the last point, I think the literature on represented spaces will be useful. There is a lot written about this; I recommend this recent survey of Schroeder (and the sources in its bibliography). Granted, this is somewhat more geared towards computable analysis than (effective) descriptive set theory, but there is still strong overlap.

For the last point, besides texts explicitly on computability-theoretic descriptive set theory (e.g. the hard-to-find Mansfield-Weitkampf, the freely-accessible section $3$ of Moschovakis' book, or these brief notes of Hjorth) I think the literature on represented spaces will be useful. There is a lot written about this; I recommend this recent survey of Schroeder (and the sources in its bibliography). Granted, this is somewhat more geared towards computable analysis than (effective) descriptive set theory, but there is still strong overlap.

That said ...

I think to a large extent there's nothing to do but grab the bull by the horns. Certainly not all expositions of forcing are equally clear, but in my experience ones which attempt a significantly simplified presentation (e.g. this "analytic" approach of Scott) don't really serve well as introductions to the fully general theory. The two standard approaches to forcing are via posets and Boolean-valued models, and I would just pick one and study it directly. In my experience the poset-based approach is far simpler, and it's what I learned; however, the Boolean-valued approach avoids any need to talk about generic objects directly, which may be a significant intuitive benefit if one is uncomfortable with model theory.

For the last point, I think the literature on represented spaces will be useful. There is a lot written about this; I recommend this recent survey of Schroeder (and the sources in its bibliography). Granted, this is somewhat more geared towards computable analysis than (recursive) descriptive set theory, but there is still strong overlap.

For the last point, I think the literature on represented spaces will be useful. There is a lot written about this; I recommend this recent survey of Schroeder (and the sources in its bibliography). Granted, this is somewhat more geared towards computable analysis than (effective) descriptive set theory, but there is still strong overlap.

Roughly speaking, suppose we have a topological space $\mathcal{X}$ equipped with a canonical basis. Each point in $\mathcal{X}$ can be described by a set of basis elements, and examples like $\mathbb{R}, 2^\mathbb{N}$, and $\mathbb{N}^\mathbb{N}$ suggest that with "naturally-occuring" spaces it is often the case that such representations yield a good computability theory (we have notions of "computable elements" of each of those spaces). Polish spaces arise as a natural generalization of this setting; note that completeness can be thought of as the analogue of "every infinite sequence of decimal digits corresponds to a real number." Of course the situation is a bit more complicated than this, as the theory of represented spaces indicates, but this is still a good starting point.

Moreover, since all "nontrivial" Polish spaces are Borel-isomorphic, we get a general independence principle for "high-level" descriptive set theoretic questions. This helps motivate them on the classical side.