Luther was really upset by Leo X. The entirety of Luther's Treatise on the Power and Primacy of the Pope could be summed up as, "Leo, you're overreaching your authority. Stop it, for Christ's sake."
I appeal to you therefore, brothers and sisters, by the mercies of God, to present your bodies as a living sacrifice, holy and acceptable to God, which is your spiritual worship.
Verse 1 is partly quoted in Apology of the Augsburg Confession, Article 12: Repentance (AP 12.132) along with several other passages to show that repentance is not about working off our sins to satisfy God, but about a total change of the person that leads to a different way of living.
Verse 1 also comes up a bit later in Article 15: Human Traditions in the Church (AP 15.45) as part of Melanchthon making a delineation between the "true death" that leads to "the spiritual exercises of fear and faith" that only happens through the cross and other "voluntary and necessary" such as fasting that serve to control the desires of the flesh but do not and cannot justify (AP 15.46-48).
Verse 1 is quoted again in Article 24: The Mass (AP 24.26) where Melanchthon explicitly spells out that, "'spiritual worship' refers to worship where God is recognized and is grasped by the mind, as happens when it fears and trusts God. Therefore, it is contrasted not only to Levitical worship... but with any worship in which people imagine that they are offering God a work ex opere operato." There is a key difference in attitude. Either you're worshiping God because of all that God does for us, or you're worshiping God because you think it allows you to indulge in sin elsewhere.
Verse 1 is called out by editorial insert a bit later in the same article (AP 24.88) under the alternate translation of "spiritual worship" as "reasonable service" near the end of a line of argument in which Melanchthon turns to the Orthodox or Greek service to point out that what is done in the mass is not a re-sacrificing of Jesus but an offering of praise to God, a reasonable service.
Verse 2 gets a nod in Formula of Concord, Solid Declaration, Article 6: Third Use of the Law (SD 6.12) as the argument is made for the role of the Ten Commandments for the reborn Christian. Once reborn, the Spirit, which activates the faith in the individual, then guides the individual with the Ten Commandments so the reborn Christian might learn what the acceptable will of God is.
Verse 5 is cited in Apology of the Augsburg Confession, Article 10: The Holy Supper (AP 10.3) through an editorial insert within a quote from Cyril of Alexandria reflects on the connection Christians have with Jesus and each other in communion--a point of agreement between the Reformers, Rome, and the Orthodox.
Verses 6-8 are cited in a footnote in Smalcald Articles, Part 2, Article 4 (SA 2.4.9, n. 59) to a parenthetical comment of Luther's noting that not all bishops are equal in gifts, even if they are equal according to their office. I'm not sure if this footnote is Luther's or an editor's, but given Luther's usual use of parenthetical jabs, I'm guessing it was not meant as a scriptural allusion.
This pericope gets some significant play in Luther's Treatise on the Power and Primacy of the Pope, but not exhaustively, so I'll be presenting these citations in the order they come up in The Book of Concord. Here are the particular verses cited:
[Jesus] said to [the disciples], “But who do you say that I am?”... [Jesus said,] "And I tell you, you are Peter, and on this rock I will build my church, and the gates of Hades will not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”
--Matthew 16:15, 18-19
Verse 18 is the first citation in the BoC, and it comes up in Apology of the Augsburg Confession, Article 4: Justification (AP 4.85) as an editorial insert in a place where Melanchthon alludes to the consolation of faith in God's forgiveness of sins because of Jesus--consolation so firm that not even the gates of Hell can overcome it. The same argument is made a bit later in Article 24: The Mass (AP 24.12) with verse 18 again being cited through editorail insertion.
Verse 19 is cited in a footnote to Smalcald Articles, Part 3, Article 7: Concerning the Keys (SA 3.7.1, n. 40) along with Matthew 18:18 and John 20:23 as places where it has been recorded that Jesus gave the church authority to bind and loose sins.
Turning now to Luther's Treatise on the Power and Primacy of the Pope, verses 15, 18, and 19 all get quoted in rapid succession. Verses 18 and 19 are both quoted (Tr 22) as evidence from Rome in defense of papal primacy. Luther, doing some textual unpacking, looks to verse 15 to show that the question is not, "Who do you, Peter, say that I am?" but "Who do y'all, my disciples, say that I am?" (Tr 23) and then shows that verse 19, which is directed to Peter, is repeated later in Matthew 18:18 as applying to all of the disciples. Luther then addresses directly the claim that verse 18, "on this rock," means Peter by citing Origen, Ambrose, Cyprian, Hilary, Bede, and Chrysostom to argue that the traditional translation is not on Peter the church will be built, but on this confession of faith the church will be built (Tr 25-29). Verse 19 comes up again as Luther argues that the pope has gone beyond the command of Jesus, overreaching the authority to bind and loose sins and claiming temporal and worldly power (Tr36). Finally, verse 19 is cited (Tr 40) as one of the ways papal authority has been abused in such a way that fits part of Paul's description of the Antichrist in 2 Thessalonians 2:4 before claiming the rest of those markers is also true.
Verse 19 is also cited in a footnote in The Small Catechism's "How simple people are taught to confess" (SC 4.28, n. 95) as it was in Smalcald Articles above, a citation to show Jesus' command and authorizing.
Verse 18 is cited in Formula of Concord, Article 11: Election in both the Epitome (Ep 11.5) and the Solid Declaration (SD 11.8, 50) to consistently make the point that God's eternal election will not be shaken even by the gates of hell.
Faith and service. The order does matter for Lutherans. Service alone leads to faith in something or someone other than Jesus, while faith leads us into service because of Jesus by the power of the Holy Spirit. There's also an undercurrent here, because of the Romans 12 passage, of the importance of community. Luther's argument for a more egalitarian leadership of the Church isn't just because he didn't like what Leo was doing. Luther's claim is that the leader of the Church is Jesus, and Jesus alone, so no one else should been seen as the leader of the Church. Indeed, we need other Christians to help us remain faithful.