According to Protestant interpretations, is Romans 3:5-6 a text relating to Theodicy or something else?

A dictionary definition of theodicy is: "Vindication of divine providence in view of existence of evil" (The Concise Oxford Dictionary, 1976 edition). You ask if Romans 3:5-6 relates to theodicy, or to something else.

It does relate to theodicy; also to other things, the big clue being "But", which begins verse 5. The need is to go back to where Paul's point starts to be made (in my Protestant view) from chapter 1 verse 16, his argument on that point being concluded at chapter 8 verse 1. It is therefore necessary to study his entire argument in order to understand the depths of the various points he is making. This is one instance where only examining two verses out of a total of 170 just will not suffice.

"But" - what? "But if our unrighteousness commend the righteousness of God, what shall we say?" This means that the argument for the righteousness of God has already been made (prior to which the argument for humanity's unrighteousness had been undeniably established.) Our unrighteousness, when compared with God's righteousness, shows up the commendable righteousness of God. In light of that fact, "what shall we say?"

Paul is aghast that anyone would therefore conclude that, when God takes vengeance, he is being unrighteous. He warned that some opponents have slanderously accused Christians of saying it's okay to do evil when that glorifies God's righteousness, by contrast. No, that is what those who do not understand the righteousness of God might think. And who are those particular ones, at that particular time of writing? Why, the ones Paul had previously been writing about, those Jewish people who thought circumcision was the critically important thing that identified a righteous, law-keeping people. He has already blown that one out of the water, to make us face the fact that God judges both Jew and Gentile impartially; that although the Jews first heard the gospel of Christ, which is God's power unto salvation, the Gentiles equally get to hear it, and it is faith in that gospel which saves all who believe.

Continue reading and note Romans 5:2, "Therefore being justified by faith, we have peace with God through our Lord, Jesus Christ: by whom also we have access by faith into this grace wherein we stand, and rejoice in hope of the glory of God." Then verses 19 to 21 add that by Christ's obedience, "shall many be made righteous. Moreover the law entered, that the offence might abound. But where sin abounded, grace did much more abound: that as sin hath reigned unto death, even so might grace reign through righteousness unto eternal life by Jesus Christ our Lord."

There are the contrasts again. Despite our sin, God has graciously provided salvation to those who gain access by faith of Christ, to have all their unrighteousness contained in the hand of God. This shows the righteousness of God in providing such immense pardon, so that those who prefer to work for their own righteousness, or to deny that God is righteous at all, will be left to their own efforts, their own reasoning. Thus, when their judgment comes, it will be a righteous judgment.

You put the question another way: "does His wrath vindicate why He allows evil?" It is God's long-suffering and patience in allowing evil people to continue in their preferred course, that vindicates God, and leaves evil-doers without a leg to stand on, come the day of judgment, when everything hidden will be revealed. God is righteously wrathful with the ungodly, with unrepentant sinners. That is why the last book of the Bible foretells increasing judgments from heaven being poured out on an increasingly ungodly world, culminating in seven last plagues just before the day of judgment. The point of those plagues is to waken people up to their need to hear the gospel, so that they will "Fear God, and give glory to him; for the hour of his judgment is come, and worship him that made heaven, and earth, and the sea, and the fountains of waters" (Revelation 14:6-7). But the account repeats that men do not repent, nor give God glory, but curse him all the more-so. (Rev. 9:20-21 & 16:9-11 & 21) Those are the ones who prefer their own idea of righteousness to God's, those who deny God and who do not repent in the only acceptable way to God - by hearing the gospel and putting faith in Christ.