Not being a Greek scholar, how could I check this myself?

As someone who's not a Greek scholar, can read Greek, has any prior insight on this debate or is a trained historian or theologian, I'm going to narrowly address this.

First, find the actual Greek text by looking at one of the earlier available Greek new testaments, like the Novum Instrumentum by Erasmus. You'll find 1 Corinthians 6:9 as

η ουκ οιδατε οτι αδικοι βασιλειαν θεου ου κληρονομησουσιν μη πλανασθε ουτε πορνοι ουτε ειδωλολατραι ουτε μοιχοι ουτε μαλακοι ουτε αρσενοκοιται

Easy enough to narrow the word you're interested in down to "αρσενοκοιται", which is the nominative plural of "ἀρσενοκοίτης", which dictionaries give as "a male who has sexual intercourse with males, especially in a penetrative role".

But it turns out that there's been a long-standing debate about how to translate αρσενοκοιται, so we have a whole body of literature to look at.

You could obviously read everything that's been written on the subject if you were so inclined, but usually my first step is to find a literature review if one exists. There's one here, "THE SOURCE AND NT MEANING OF ARSENOKOITAI, WITH IMPLICATIONS FOR CHRISTIAN ETHICS AND MINISTRY", James B. DeYoung, TMSJ 3/2 (Fall 1992) 191-215

It's important to note the bias of the author, who in this case is a conservative professor arguing for a broad interpretation of the term, that is, one condemning homosexuality as a whole in both orientation and sexual acts, as opposed to the narrower interpretations of it referring to either just the physical act of male sex rather than homosexual orientation as a whole or even just to male prostitution.

But he does give what appears to be a lengthy and fair summary of the case for the more narrow interpretations (the term is not used by pagan writers, for example, and the verse is generally not included in the discussions of early church fathers about homosexuality), so I'd consider it a decent jumping off point to learn about the different facets of the debate and the relevant papers.

Then next steps, if you're not satisfied at that point, could involve reading the individual papers reviewed and summarized in this journal article, and looking into what if any progress the literature has made since this review was published in 1992 (although as far as I can tell from a cursory overview the debate has largely died down and most newer takes from the internet era just seem to be people quoting the Boswell/Scroggs/Wright papers from the 70s and 80s discussed in the review without adding much insight of their own.)

I hope this helps as an example for how you can dig into a topic and familiarize yourself with the relevant literature pretty quickly without any previous expertise, but I'll leave giving personal opinions on the topic to other commenters here who were familiar with the debate more than 20 minutes ago, I'm not that presumptuous.

This will not be an answer to your questions of historical substance but rather applying to your third, that of access to Weigle's correspondence.

The collection is currently held in the Manuscripts and Archives Collection of the Yale University Library. The collection is, as far as I am able to determine, undigitized and only correspondence with other members of the RSV Committee, as well as meeting notes, appear to even be held on microfilm. However, it appears that you might be able to have the library digitize the relevant correspondence for you, possibly for free.

The finding guide for the Luther Allan Weigle papers can be found here and the Yale University Libraries Digitization Service information can be found here. If you have a citation for the student's letter, ideally with a name of the student or a folder and box number, you should be able to identify the relevant document in the finding guide. At the moment, digitization requests are limited to 4 folders for unbound documents. However, orders for document scans are free at this time to pilot a new service to support digitization of Special Collections.

If you are unable to find a precise note of who wrote the letter, or where it is in the collection, you would need to access the documents in person. You could do this yourself by traveling to New Haven. While the collection is listed as open for research, and thus able to accessed in theory by an outside researcher registered with the university, a process detailed at this link, the libraries overall are not accessible to outside researchers this term due to COVID restrictions. If traveling to New Haven in the future is infeasible, you can also hire a Yale-affiliated researcher to go into the archives for you. There is no centralized system to do so, but the Manuscripts and Archives Collection does provide a good guide of the ways you might approach hiring a researcher and what you must take into consideration when doing so. This guide is in some ways specific to accessing collections in the YUL Manuscripts and Archives depository but the general steps of locating the finding guide, identifying the location of documents within archive, and researching library policies on digitization, public access, and research hiring are useful for assessing access possibilities in most archives.

If we’re talking about a (re)translation of this contentious term as "sexual perverts," this would be a truly mystifying modification. Nothing in the word's form/etymology, nor in the evidence of any of its actual usage, points in the direction of a broader sense of “(sexual) perversion” that this term/phrase usually conjures. A better analogous fit to that is another Greek word, porneia, usually understood with a broader sense of “sexual immorality.” [Note: see now my addendums #3 and #4 at the end of this post, about RSV’s “sexual perverts.”]

In any case, it’s more or less universally agreed by scholars that the inspiration for the actual word in question was likely the Septuagint (LXX) translation of Leviticus — coined as a neologism that brought together two of the terms used in LXX Leviticus' prohibition against males sleeping with (or "bedding") other males. This is also paralleled by a similar prohibition in the apocryphal early Jewish/Christian Sibylline Oracles, which may in fact be alluding to the same prohibition from Leviticus, except using a different synonym for "bed" (this one from the verb εὐνάω). In modernity, the connection with LXX Leviticus was noted as early as 1870, in the lexical entry for the word in E. A. Sophocles’ Greek Lexicon of the Roman and Byzantine Periods; and probably earlier, too, if I had to guess.

BDAG, the premiere contemporary academic lexicon of New Testament Greek, notes the Leviticus connection, and glosses the word itself as "a male who engages in sexual activity w. a pers. of his own sex, pederast." It also compares the compound term to μητροκοίτης, mēterokoitēs, “one who has intercourse w. his mother." I’ll add to this that there are other similarly formed words, too, like ἀδελφοκοιτία, adelphokoitia — “having sex with siblings” —, or δουλοκοίτης, doulokoitēs, “one who has sex with a slave,” or the verb ἀνδροκοιτέω, androkoiteō, used for example to refer to a woman who had sex with a man while pregnant.

Also note that BDAG’s definition “...who engages in sexual activity with...” is a little euphemistic. In ancient terms, the referenced sexual activity would almost exclusively suggest one thing: actual anal penetration.

In any case, its connection with the anal penetration of another male — whether this was thought to take place in relation to pederasty, or solicitation of male prostitution, or what we could call doulokoitia, or in broader terms — was so well-established that the term seems to have undergone some later evolution to actually also denote "anal sex" in general (even in heterosexual contexts!), as attested to in the sixth century John Nesteutes.

A recent examination of the terms in at least some authors/theologians in the early church can be found in John Cook's "μαλακοί and ἀρσενοκοῖται: In Defence of Tertullian’s Translation,” in the journal New Testament Studies.

[Edit:] I'm surprised that the original letter referred to in the OP doesn't seem to have made its way online, as far as I can tell. I can only find one short excerpt from it:

I write this letter after many months of serious thought and hard work, partly to point out that which to me is a serious weakness in translation, but more because of my deep concern for those who are wronged and slandered by the incorrect usage of this word. Since this is a Holy Book of Scripture sacred to the Christians, I am the more deeply concerned because well meaning and sincere, but misinformed and misguided people (those among the clergy not excluded) may use this Revised Standard Version as a sacred weapon, not in fact for the purification of the church, but in fact for injustice against a defenseless minority group which includes the sincere, convicted spiritually reborn Christian who has discovered himself to be of homosexual inclination from the time of his memory. I write this letter with certain homosexual individuals in mind - Christians who would die for their faith, their church, and their lord but who cannot alter their biological state of being.

I hope that the committee responsible for considering any possible corrections or revisions of the R.S.V. text may take my case here presented into consideration

But the entirety of the "case here presented" seems to be missing, along with the original response from Weigle. In fact, honestly, I find it a little suspicious that this letter seems to have played such a pivotal role in things like this, and yet... doesn't seem to be published anywhere.

[Edit 2:] Ah, I think I see why: one of the main persons behind the aforementioned documentary seems to be withholding it in order to build up support for her new book. I've now messaged her asking why she won't let the rest of us access such historically-important documents that she clearly has at least photos of. Yeah, we'll see how that goes over.

[Edit 3:] Okay, so I found a post by the documentarian behind the 1946 movie, who's responsible for publicizing this whole thing; and in it she writes that

In the 1971 revision [of the Revised Standard Version], another team used “sexual pervert” in lieu of “homosexual,” but by then, in the culture, “homosexual” and “sexual pervert” had become synonymous.

So I suppose she could be correct that in the early 70s, somehow "sexual pervert" would denote "homosexual" in particular to people. But then why revise it at all, if the terms would have been considered roughly synonymous?

[Edit 4:] I'm now realizing that "sexual perverts" in RSV's 1971 revision actually translates two paired terms in 1 Corinthians 6:9, malakoi and arsenokoitai — not just the latter term, as implied elsewhere. So I suppose that makes marginally more sense, if (as is the case) both terms refer to those who engaged in what were thought to be illicit sex acts, and were just rolled into one broader category by the translators. ESV seems to have taken the same route, but rolling them into the more specific category of “men who practice homosexuality.”

While we're waiting for a strong answer here, you may be interested in a post on this matter from r/AcademicBiblical, by u/koine_lingua, which goes into detail on the original Greek word in question (ἀρσενοκοῖται):

https://www.reddit.com/r/AcademicBiblical/comments/1i3wiu/pauls_terms_for_homosexual_practice_in_1/?ref=share&ref_source=link

I'm a Classicist, and I'm still early in my career, but I've focused some of my study on ancient sexual practices, New Testament Greek, Judaism in the context of the Ancient Near East, and Early Christianity.

Most of this question is outside of the purview of history, but some of the responses trying to discuss the non-historical aspects are severely lacking in nuance. There is a considerable amount of debate surrounding this word, and a lot of the "clear reading" of the word, at least in an English context, is deeply influenced by a longstanding sociocultural bias against homosexuality. Paul's first letter to the Corinthians is nearly 2000 years old- attempting to parse out the word with an understanding of modern Greek linguistics or through the lens of the recent (last several centuries) English understanding of the word is quite silly. You can claim an etymology of the words from which the term is derived doesn't point to a translation of "sexual pervert", but words don't exist in isolation of their constituent parts - if you're a linguist studying ancient words, you parse their semantic range by the manner in which they're used and then transfer the word into our context, pulling in an understanding that links the old to the new.

The current debate isn't so much "Does this word give the connotation of one male human person having sexual relations with another male human person?", but instead, "In light of the word's ambiguity and our knowledge of ancient sexual practices, to what acts in particular was this referring?" With that in mind, there's a grain of truth to "Homosexuality" first being introduced into an English translation of the New Testament in the mid 20th century, in so far a word for the internal persuasion of sexuality didn't exist until the 1860s, when it was coined by a German psychologist. As that notion would have been wholly foreign to an ancient, if we're granting the notion that it's referring to homosexual behavior, then a more proper translation becomes "They who act as penetrator in fulfillment of their homosexual urges".

This is where the debate comes in. It's quite common to argue that, per the social mores of ancient Greco-Rome (and some argue this for the context of Leviticus via sexual practices of the Ancient Near East as well, if arsenokoitai is, in fact, pulled from Leviticus), that the sexual practices involving two "men" of which Paul would have been aware would have likely occurred in the context of the rape of an enslaved person or the pursuance of a young eromenos, such as with Ancient Greek pederasty. Homosexual, then, would be an inappropriate translation in so far modern sexual ethics are as such that, in a manner unfamiliar to the era of Paul, two gay men will pursue one another romantically or sexually in a manner concerned with consent, including the consent one can only grant as coming of the proper age. As such, you could translate the word as "sexual pervert" to give a translation as meaningful to the modern reader as would have been the case for the church in Corinth. There are those who dispute this, but that's the TL;DR version of the argument.

Further confusion is introduced as one ancient commentary includes it in a list of economic sins, and there's at least one instance of it being said a husband committed the sin with his wife. In the context of the last 500 years, further, French and German translations have often treated the word quite differently than we have in English, usually giving the indication of pederast or one who sleeps with catamites. I'm not really answering your question per the terms of the field of history, so I'm not worrying about giving sources. I just hope this is enough of a "Huh. That's more complex than I thought it would be" to propel further research.

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