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.
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.
However, I don't really think that (set-theoretic) forcing and inner models are best approached via computability theory, even for a computability theorist; while they do have computability-theoretic aspects, focusing too much on this will obscure the fundamentally set-theoretic nature of the topic which is necessary to any deep understanding. In particular, while there is a notion of forcing in classical computability theory, in my opinion it's far too weak to yield a good bridge to set-theoretic forcing.
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 general set-theoretic forcing and inner models, I think the right starting point (overlooking the concern of the paragraph above) is hyperarithmetic theory. Not only is this a "lightface" analogue of the Borel hierarchy (so it's relevant to the previous point too), it is tightly connected to the construction of $L$: in a very strong sense, $L$ is what you get when you "continue the hyperarithmetic hierarchy through the ordinals" (the relevant term here is "master codes" - see here).
Hyperarithmetic theory is also useful for understanding set-theoretic forcing. Forcing in classical computability theory (e.g. Jockusch-Soare forcing) is a bit too simple to provide a good starting point for set-theoretic forcing in my opinion. One wants to shift from classical computability theoretic forcing to forcing over some "set theory flavored" structure. It turns out that forcing over levels of $L$ is a very good context for doing this. For example, consider the proof that a comeager set of reals $x$ satisfy $\omega_1^{CK}(x)=\omega_1^{CK}$; this is basically "Cohen forcing over $L_{\omega_1^{CK}}$." In general, there's a rich theory of forcing over admissible sets.
My personal favorite treatment of the hyperarithmetic hierarchy and related notions is (the first few parts of) Sacks' book Higher recursion theory (it should be available via ProjectEuclid, but that link doesn't work at the moment).
Now granted, in a precise sense we can make sense of forcing "computably" rather than "hyperarithmetically;" however, while it's a very nice result I don't think this is a good way to approach the subject as a beginner from computability theory.
One final topic worth mentioning re: forcing, specifically connected with combinatorial topics like $\mathsf{CH}$, is the theory of effective cardinal characteristics of the continuum. This amounts to a "computable analogue" of the analysis of the interval $[\aleph_1,2^{\aleph_0}]$ via forcing, with set-theoretic forcing notions usually having corresponding computability-theoretic ones which yield analogous results. For example, we have a direct connection between the following two facts:
It is (relatively) consistent with $\mathsf{ZFC}$ that there is a cardinal $\kappa$ such that $(i)$ there is a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that every $g:\mathbb{N}\rightarrow\mathbb{N}$ is escaped by some $f\in F$, but $(ii)$ there is a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that every $g:\mathbb{N}\rightarrow\mathbb{N}$ is dominated by some $f\in F$.
There are hyperimmune degrees (= compute a function escaping all computable functions) which are not high (= compute a function dominating all computable functions).
Note that a consistency result is replaced by an outright fact.
Here the idea is to shift from cardinality on the set-theoretic side to lowness/highness notions on the computability-theoretic side. Moreover, while the applications generally go from the set-theoretic version to the computability-theoretic version, there is at least one case where we get a result on the set-theoretic side building on an initial result on the computability-theoretic side; the set-theoretic result (which was much harder) was gotten by Ivan Ongay-Valverde, and the computability-theoretic result is joint between Ivan and myself.
However, I think that this is a bit of a double-edged sword: while much more accessible to the classical computability theorist than hyperarithmetic theory, the conceptual shift in the previous sentence does lose a lot of the set-theoretic "spirit." I'm not sure how much I recommend it as a pedagogical starting point.
