Apparently the Theory is not from Analysis of Pre-Extant Texts
If C. Marvin Pate is correct in his Romans commentary statements (no page numbers shown in that Google Book link, but it is under the section where he discusses those verses in the commentary), then the two main reasons this becomes a question at all for this passage has nothing to do with any earlier extant texts that demonstrate such is the case. That is, there appears to be no earlier text showing these statements that is known to exist from which to do any direct analysis from. Pate states it so (bold added by me):
The theory is that 3:21-24 is Pauline, but 3:25-26a is pre-Pauline. Those who support this view point to two "un-Pauline" ideas in these verses: (1) the sacrificial language of 3:25, especially "faith in his blood"; (2) God overlooked past sins committed in the Old Testament. The upshot of these arguments is that Paul corrects a hymn that emphasized God's making sinners right with himself by introducing the notion that the death of Christ did more than that: it appeased the wrath of God toward humankind.[66] The crux of the matter is that Paul introduces the judging aspect of divine righteousness into a creed that emphasized only the saving aspect of God's righteousness. These arguments appear to be valid, bit if one accepts Ephesians 1:7 as Pauline and the report of Acts 20:28 as reliable, then referring to Jesus' death as a "bloody" sacrifice is not un-Pauline at all. Furthermore, the fact that only once in his letters does Paul mention God's passing over the sins of the past need not preclude this mention as being from Paul. But most important, already in Romans 1:17-18 (which no one things is pre-Pauline) we see Paul juxtaposing God's saving and judging righteousness. Actually, for Paul, both are needed to present the character of God, who is both holy and loving (so 3:26b). There is no need, therefore, to resort to the theory of Romans 3:25-26a being pre-Pauline.
So if Pate is accurate, the support of the theory appears to largely be based upon preconceived notions of what Paul would say (and that from a perspective that appears to not consider the text as being inspired by God if that is important to you or not; it is to me), rather than any actual textual evidence showing an earlier tradition from which he might be drawing.
Pate's answer, however, does not really address why it is deemed that the earlier source might have been a hymn rather than some other type of literature. That is, he does not address why that genre of literature is believed to be backing Romans 3:25-26 rather than some other genre.
Four (Three?) Points of Argument for Non-Pauline
With no existing earlier text to prove the point, other means are attempted. So to answer your question of:
What sorts of analyses are used to determine whether part of all of
this text existed in some form prior to Paul’s writing?
Perhaps only the analysis of Paul's thoughts in relation to the particular framework those analyzing hold to, which apparently does not allow for something other than what they conceive as potentially being Pauline as being so.
This appears to be the case when you examine information from Charles H. Talbert's article "A Non-Pauline Fragment at Romans 3:24-26?" (Journal of Biblical Literature 85:3 [Sept 1966]: 287-296).
Talbert believes "beyond reasonable doubt" that there is an insertion (287). However, his paper is not specifically about arguments proving insertion, but rather that v.24 should be excluded, and only v.25-26 included as being the traditional part. Nevertheless, he does summarize the arguments used by "Bultmann, Käsemann, and Hunter" (who hold v.24-25 are the insertion, not v.26). There are four he notes on pages 287-288 (I will offer my own thoughts in italics following the main summary of each):
- Linguistic/Syntactical: which relies on (1) "certain terms, like ἱλαστήριον, πάρεσις, and προγίνεσθαι occur nowhere else in Paul," (2) "the awkwardness of the opening participle δικαιούμενοι in 3:24" (287). Talbert himself will argue against (2), which he notes is the only evidence given by the others that v.24 should be part of the inclusion, all the rest residing in v.25 (287). Regarding (1), I believe it is a grave injustice to a writer to presuppose a limitation on the vocabulary they might use to convey their thoughts; doubly so if one believes there is also a Divine author behind the text as well. This argument would hold no weight for me. Regarding (2), I agree with Talbert that there are readings that do not see this as awkward.
- Stylistic: "it is argued that the concentration of prepositional phrases and genitival constructions ... is often a mark of a liturgical style." Of this evidence, Talbert himself admits it is "in themselves not a strong argument" but "can be used as supporting evidence." Regarding style, however, Talbert believes v.24-25 fail to be a "balanced structure," unlike what he will argue for v.25-26 (288). This is the only reference so far that I have found so far that in any way points to "hymn" as genre, and that only obliquely through the mention of liturgy (which could be more than hymn). Probably it is more discussed by Hunter perhaps, who Talbert footnotes as the one making the stylistic argument. For myself, I would agree with Talbert that the argument is weak, as there are certainly other reasons besides liturgy for finding such concentrations.
- Theological: "there are a number of theological concepts in these verses that are not Pauline," noting (1) "Jesus' death as a ἱλαστήριον," (2) Paul's emphasis on "cross" not "blood" of Christ, (3) God's righteousness "demanding expiation for past sins," (4) "in vss. 24-25 the new covenant is described as God's gracious restoration of the old covenant" (288). Another grave injustice to a writer is to presuppose what is their "theological concept," ignoring the writing as it is, and then impose that as a parameter for excluding the statements as original. It is like me presupposing that the OP did not write "I recently ran across the hypothesis," and that really some later editor is the one who ran across it and edited that into the question. Additionally, it again assumes that there is no Divine thought behind the text also, which would have consistency with other topics/ideas, perhaps expounded more by other writers of Scripture. Regarding (1)-(3), the OT is enough background to argue that. Regarding (4), that is purely false in my view; the assertion assumes the view that God's righteousness refers to OT covenant faithfulness (as New Perspective on Paul advocates argue), which I believe to be a false understanding of what God's righteousness refers to, namely that He defines and does right. So in this passage, it would not have been right for God to pass over those sins (because He had said they would be judged) without having a ground (Christ's planned death) to pass them over.
- Contextual: The v.25 notation that "God had passed over sins does not fit," primarily in light of "Rom 1:18ff. Paul says that God's wrath characterizes the past and past sins." Supposedly the insertion of v.24-25 "resolve" this contradiction in their mind (288). Talbert himself, in arguing for v.25-26 as inclusion, and that v.24 is original, states that his argument showing parallels in v.25 and 26 then "destroys Käsemann's argument from the context" (291 n.16). That is, it nullifies this fourth argument. For me, their resolution of the contradiction is not really a resolution, but rather affirming there is a contradiction, but only introduced because Paul borrowed something that he did not fully fit into his theology (or a later editor added without thinking). I agree with Talbert that the phrase does fit the context (though perhaps for somewhat different reasons). I would also argue that Rom 1:18ff shows God's passing over sins, by those individuals being allowed to devolve in their sin, giving time for repentance, rather than being immediately put to death.
In light of Talbert's paper, Pate's summary shows that his point (1) equates in part to both points (1) and (3) here, and his point (2) with (4). It also demonstrates to me that my critique from Pate's summary is also accurate. It is primarily the presuppositional framework from which the question arises at all.
Basically, since they...
- Do not believe Paul (and God) would use certain words,
- Do not believe Paul (and God) would concentrate prepositions/genitives himself,
- Do not believe Paul (and God) would convey certain theological concepts, and
- Do believe Paul (and God) have contradicted themselves in context
...then they feel it is "un-Pauline" and thus must not be original. Yet in the first three cases, they are presupposing such is already "un-Pauline" to make the argument that it is "un-Pauline," and so those three "evidences" are begging the question (assuming the conclusion).
The remainder of Talbert's arguments for v.25-26 instead of v.24-25 to me only prove one thing, that the verses are likely a parenthetical thought. But unlike his conclusions that lead him to argue for them to be a later, non-Pauline insertion (292-296; so he is not arguing that Paul borrowed the thought, Talbert is arguing it is a later editorial insertion by someone other than Paul), I find his evidence simply indicating that Paul wanted to give a parenthetical summary of 3:10-24 in 25-26 to hammer home his rhetorical question of v.27.
General Points of Analysis Used
Talbert notes two sources in which "criteria for ... credal formulae" are presented (292 n.19). One is Ethelbert Stauffer's 12 points noted in his work New Testament Theology (pages 338-339), which I found a summary of online in The Westminster Dictionary of New Testament & Early Christian Literature & Rheotoric (page 224). This work does note that these criteria "should also be useful in detecting the presence of hymnic material." Considering that Talbert states Romans 3:25-26 "satisfies at least 2, 4, 7, 8, 9, 12" of Stauffer's criteria (292 n.19), then this 50% match to the criteria must be enough for some people to classify it as hymn.
I personally disagree that it meets criteria 2; agree 4 is met; agree 7 is met, but also would argue that anithetical or anaphoral arrangement is not exclusive to creeds/hymns; consider 8-9 rather subjective as to whether something meets that or not; and would argue 12, while perhaps true of creeds, is also true of inspired Scripture in which God is communicating norms, and so has little value in identifying an insertion in Scripture. So for me, the passage does not match all that well to the criteria anyway.
The second work is Reginald Horace Fuller's The Foundations of New Testament Christology (page 21). I have not yet analyzed those criteria.
Conclusion
To answer your title question, I would agree with Pate that there is no good reason to not consider the passage as original to Paul. The arguments to even question it are primarily circular, question begging points. Coupled with the fact that even for one holding to the view, such as Talbert, the match appears to be at best 50% of one set of criteria used to determine such things (which some of the criteria themselves have some questionable value).
