The Lévy-Montague reflection theorem, of course, explains that every first-order expressible property $\varphi$ in the language of set theory reflects from the set-theoretic universe from $V$ to some rank-initial segment $V_\alpha$, in the sense that $\varphi(x)$ is absolute between $V_\alpha$ and $V$. Indeed, for any such $\varphi$ there is a closed unbounded class $C$ of cardinals $\kappa$ such that $V_\kappa\models\varphi[a]$ if and only if $V\models\varphi[a]$, for any $\kappa\in C$ and $a\in V_\kappa$.
Since the first-order language of set theory is quite robust, capable of expressing versions of almost any mathematical idea, the conclusion is that almost every mathematical property is subject to the reflection phenomenon. On this view, reflection is a fundamental feature of mathematical reality.
Let me mention a few further interesting things about reflection.
A cardinal $\kappa$ is $\Sigma_n$-correct if all $\Sigma_n$ assertions are absolute between $V$ and $V_\kappa$. These are the cardinals to which assertions of that complexity are reflected. There is a closed unbounded class $C^{(n)}$ of $\Sigma_n$ correct cardinals, also known as the $C^{(n)}$-cardinals. But beware, one cannot necessarily intersect these class clubs to form $\bigcap_n C^{(n)}$, the class of fully correct cardinals, since the definition of being $\Sigma_n$ is not uniform in $n$, but increasingly complex as $n$ increases.
Reflection is equivalent to the axiom of replacement over the other axioms of set theory. This is because if $A$ is a set and you have witness objects $b$ for every element $a\in A$, then by reflection there is some $V_\alpha$ in which those witnesses reside and for which the witnessing property is absolute. So now the set of witnesses can be picked out via separation as a subset of $V_\alpha$.
Reflection is also equivalent to the principle of transfinite recursion. I explain on my blog.
Philosophical ideas based on reflection are often mentioned, by some of the authors you mention in your post, as providing philosophical justification for the existence of large cardinals. For example, one early idea is that if we accept reflection as a fundamental principle, then ZFC is saying essentially that the class of ordinals is a regular strong limit, and so we should expect there to be some rank initial segment $V_\kappa$ in which this is true, making $\kappa$ an inaccessible cardinal. Further arguments continue with this to much larger large cardinals. See work of Reinhardt, and also Maddy on Believing the Axioms.
In my paper, Categorical large cardinals and the tension between categoricity and set-theoretic reflection, J.D. Hamkins and H.R. Solberg, we discuss the inherent tension between reflection and categoricity in set-theoretic foundations. Mathematicians often prefer categorical accounts of their basic mathematical structures, and yet also prefer reflection, which is at heart anti-categorical.
You seem to inquire also about extensions of reflection, and so let me mention some broader contexts in which reflection is considered.
Weak contexts. It is natural to inquire about reflection not just in ZFC contexts, but in weak contexts such as ZFC-, that is, without the power set axiom. Here, the $V_\alpha$ hierarchy might not exist as sets, and so one will not be wanting to reflect to $V_\alpha$, but rather to some transitive set. That is, the reflection principle would say that every $\varphi$ is absolute from the ambient universe to some transitive set. It was an open question whether this was provable in ZFC-, and I think this was resolved by Gitman and Friedman, but I shall look up the reference.
Stationary reflection. A fruitful notion in set theory has been the notion of stationary reflection, which holds of a given stationary set $S\subseteq\kappa$ when there is some initial segment $S\cap\gamma$ that is stationary in $\gamma$. When $\kappa$ is a sufficient large cardinal, such as weakly compact, then every stationary set reflects in this way, and so the existence of a stationary nonreflecting set is an anti-large cardinal property.
Second-order reflection. It is also natural to inquire about reflection of second-order assertions. Here there are several ways to proceed.
One natural initial meaning of second-order reflection would be that any first-order assertion $\varphi(X)$ about a second-order class $X$ reflects to some $\langle V_\kappa,\in,X\cap V_\kappa\rangle$. This is the kind of reflection occuring in the statement ``Ord is Mahlo'', where one asserts that every definable closed unbounded class of ordinals contains a regular cardinal. This enables reflection to inaccessible cardinals, and can be seen as a robust strengthening of the Zermelo-Grothendieck universe axiom.
This kind of reflection at a cardinal $\kappa$ provides a characterization of the weakly compact cardinals $\kappa$, those for which every $\Pi^1_1$ class reflects from $V_\kappa$ to some smaller $V_\gamma$.
A stronger kind of reflection will consider more complex assertions about classes, or even second-order properties themselves. In Gödel-Bernays set theory GBC, for example, one could consider second-order reflection principle, stating that every second-order assertion $\varphi(X)$ about a class $X$ reflects to some $\langle V_\kappa,\in,X\cap V_\kappa\rangle$. This principle is not provable in GBC, nor even in Kelley-Morse set theory, and indeed it erases the distinction between GBC and Kelley-Morse set theory, since even GBc + reflection implies KM+CC and much more. The large cardinal strength of second-order reflection is between $\Pi^1_n$-indescribable cardinals and a measurable cardinal, or actually an $\omega$-Erdős cardinal. We have a summary discussion of this in section 7 of our paper:
Reflection with urelements. I find it natural to consider reflection principles in set theory with urelement atoms. In the case of a set of atoms, the theory is bi-interpretable with ZFC and things work as expected. When there is a proper class of atoms, however, then one should not reflect to rank-initial segments, since these are not sets, but rather to transitive sets as in ZFC-. The case of second-order reflection is quite interesting, and carries surprising large cardinal strength, particularly when there are more than Ord many atoms. For example:
Theorem. (Yao,Hamkins) GBc set theory with the abundant atom axiom plus second-order reflection implies the consistency with ZFC of a supercompact cardinal.
In fact, we identify a collection of bi-interpretation results between these urelement theories and pure set theories.
