Answer

1. Yes. It has religious status, and sometimes also halakhic status (“why do I need a verse? It is logical reasoning”). At times it also takes part in interpreting the Torah, as will be explained.

2. Moshe Halbertal wrote a book (Interpretive Revolutions in the Making) about moral considerations within the halakhic interpretation of the Sages. Yitzhak Gilat as well, in The Development of Jewish Law, shows a number of such processes that involved external considerations together with interpretive-exegetical ones.
If you look into these and other sources, I think you will discover that these two possibilities are not as dichotomous as you present them.
In my view, this kind of interpretation involves two types of considerations: the motivation is moral, but they do not simply make a change unless it has an interpretive-exegetical basis. Alternatively, sometimes the motivation is that a text calls for interpretation (for example, there is an “et” that needs to include something), and they look for a plausible exegetical outcome (or the most plausible one possible). Only when both requirements are met do they make the interpretation and derive the halakhic conclusion.
In extreme cases, the Sages have more revolutionary tools, such as nullifying something from the Torah through passive omission, “sit and do not act” (and sometimes even through positive action, as some medieval authorities noted), or instituting an ordinance. In such cases the change is made because of a moral-social motivation, and it is done only on the rabbinic level and not through reinterpretation of the core Torah-level law. But when it is done as interpretation, then apparently the motivation is not only the moral one (otherwise they would institute an ordinance or uproot the law). When it is done as interpretation, that means the interpretive tool holds water. I assume the Sages are not just playing games.
True, the interpretations sometimes seem unreasonable, and so the impression one gets is that the Sages did whatever they wanted. But it is worth noticing that this also happens where there is no visible moral or logical motivation. That is, it is not that when needed the Sages use the interpretive method arbitrarily; even when they do not need to, they do so. My conclusion is that in both cases the use is not arbitrary; rather, the methods of interpretation are probably not fully familiar to us, and so they seem strange and puzzling. I assume that if we knew them, we would understand that “an eye for an eye” is indeed interpreted according to its plain sense, but the verbal analogy of “under/under” is also compelling. The combination yields monetary compensation in place of an eye. I came to understand a bit of this in my work on the hermeneutical principles, but of course the road to a full (and critical) reconstruction of the methods of interpretation is still long. I do not mean to say that once we reconstruct them we will necessarily agree and everything will become sensible and clear. I only assume that the Sages thought this was the right way to proceed and were not merely putting on a show. It is possible that we will not agree, as in many other topics. There is nothing wrong with that, and it is even to be expected.
And beyond all that, David Henshke, in a series of three articles in HaMa’ayan 5737–38, showed that interpretation does not remove the verse from its plain meaning (there are very rare cases where interpretation contradicts the plain meaning; usually it adds something alongside it), but rather offers a parallel interpretation. Jewish law is a combination of both the interpretation and the plain meaning together (following the Vilna Gaon’s approach). Therefore, when you look at the interpretation alone, it seems as though it twists the verse, but that is a mistake. The interpretation does not offer a plain-sense explanation of the verse but a parallel explanation in addition to the plain meaning, which also remains as a halakhic interpretation. The Torah was given to be interpreted in all the ways of Pardes, all together.