Christ talks a lot about repentance, and how we must repent and believe to be saved, he asks the woman caught in adultery to sin no more (John 8:11). But if we are unable to do any good as Paul says in Romans (Romans 7:18), how can we repent of anything? And also, can we truly say we live a life of "repentance" if we will sin for the rest of our lives, sort of "the old Adam still haunting us", when in contrast we should be as "dead to sin" (i.e. sin no more)?
I hope you please allow me a little patience and latitude to answer your question, since, in fact it's a number of questions wrapped up into one post.
the Teachings of the Lutheran Church are found in the book of concord. Specifically, the Augsburg Confession has a useful article on Repentance (https://bookofconcord.org/augsburg-confession/of-repentance/)
To flesh out the historical context, For Lutherans, one quickly notices that repentance is used in a narrow and a broad way/sense: Narrow: "terrors, smiting the conscience because of sin." Broad: Both terrors smiting the conscience and faith/trust in God's promises.
It is vitally important to understand, that, from a Lutheran context, there is much overlap between repentance and faith. Just as one is unable to choose his/her way into heaven due to the lack of a free will (Ephesians. 2; 5:8; Romans 8:7; 1 Cor. 2:14, et al), so also a person is not able to make themselves repent. Repentance is not a good work we perform for God. It's a gracious gift given to us and worked in us through God's word (cf. —Acts 3:26 —Acts 5:31 —Acts 11:18 —Rom. 2:4 —2 Tim. 2:25)
"How can we repent" then? Really good question. Repentance is caused by God, not found in us. But, through water and word, Christians are 'born again'. Alongside our old, sinful self, there is a new nature. This contrast is shown in the long, beautiful section of scripture in Romans 7. For example, look at “For in my inner being I delight in God’s law” (Rom. 7:22 NIV11-GKE). The inner person "ⲉⲥⲱ ⲁⲛⲑⲣⲱⲡⲟⲛ" repents of sin and clings to God's promises. This inner/new self "lives a life of repentance" every day.
Will we "sin for the rest of our lives" then? Since we have an old, sinful nature to the moment of our death, then the answer is, "yes." But that's not the only answer. We also have a new self for whom the answer is also "no." As Paul speaks elsewhere: “For the flesh desires what is contrary to the Spirit, and the Spirit what is contrary to the flesh. They are in conflict with each other, so that you are not to do whatever you want.” (Gal. 5:17 NIV11-GKE). Notice the present tense (“ἀντίκειται” (Γαλάτας 5·17 THGNT-T)). This continues, again and again, until you die.
With this continual struggle in mind, we can answer the last question. Do we, as Christians, "put sin to death?" Luther, in the small catechism writes:
What then is the significance of such a baptism with water? Answer: It signifies that the old creature in us with all sins and evil desires is to be drowned and die through daily contrition and repentance, and on the other hand that daily a new person is to come forth and rise up to live before God in righteousness and purity forever. Where is this written? Answer:
St. Paul says in Romans 6[:4], “We were buried with Christ through baptism into death, so that, just as Christ was raised from the dead through the glory of the Father, we, too, are to walk in a new life.”
Robert Kolb and Timothy J. Wengert, eds. The Book of Concord: The Confessions of the Evangelical Lutheran Church. Accordance electronic ed. (Minneapolis: Fortress Press, 2000), 360.
Notice the paradox: We die and are raised to spiritual life—yet this happens repeatedly. That is what our life as Christians looks like. We repent of our sin and grieve over it. We rejoice that Jesus paid for the sins of the entire world (John 1:29). We rejoice that through all the power and promises in his word he delivers this salvation to me in those waters of baptism.
Let me close where I began. Your post was a network of a number of questions. I hope you will have patience with my answer, since I sought to give you a Lutheran answer to a series of questions about the Bible from official, substantive Lutheran sources.