With the first question already answered,
1. What is sin?,
Does anyone else wish to have a go at answering the other questions in the OP here?
2. What is rightouesness?
3. What is repentance?
4. What do we repent from?
5. Can we be saved while continuing in known unrepentant sin?
God bless
1. What is sin?
Answer: All which is a failure to do as we ought. It is, in a simplified way, all which is in transgression to God's commandments. In a more complicated way, as St. Paul says, "everything that does not come from faith is sin". So while the stronger brother can, with a clean conscience, eat foods sacrificed to idols in good faith and it is not a sin (since idols are nothing), the weaker brother regards such to be sinful--it is therefore a sin to violate one's conscience and do something one believes is wrong; and further it is wrong to place another in a position to violate their conscience. One man honors one day above another, but there is no difference; it is a sin to violate the conscience of one's brother, just as it is a sin to insist on one day over another as though God has commanded such a thing when He has not.
2. What is righteousness?
Answer: That which is right. That's the meaning of the word, the state or condition of rightness or being in the right; compare to the Latin derived word justice, i.e. that which is just or right. Righteousness and justice are the same thing. We know that on account of the sinful appetites that we are not righteous, but indeed we are unjust; and that we behold in God's commandment the reflection of our iniquity. For God has commanded that we love our neighbor as ourselves, and we have not done it. As Martin Luther writes, "The Law says 'Do this' and it is never done." So it is that St. Paul can say, "There is no one righteous, not even one." For there is none who is righteous, none who has done good, no one who seeks after God, we are all, without exception, wayward. And so we find that no one is righteous under the Law, for no one has kept the Law, no one has abided in God's commandments. None, except Jesus Christ. The only just one, the only righteous one. And by the grace of God His perfect righteousness, His justice, has become our justice as pure gift, through faith. Therefore, "justice is revealed from faith to faith, as it is written, 'The just shall walk by faith.'" So we have received the alien righteousness of Christ, as pure gift. For we, being sinners, have no righteousness, and no ability to obey God's good command. For all have sinned and fallen short, there is no one who is righteous, and all stand condemned under the Law as sinners.
3. What is repentance?
4. What do we repent from?
Answer: Metanoia, a change or transformation of the mind; the sorrow and grief that comes when I behold my sins for what they are; that I have sinned against God and my neighbor in thought, word, and deed; by what I have done and by what I have left undone. I have not loved God with my whole heart, I have not loved my neighbor as myself. I have gone my own way, sought myself, and pursued my own righteousness thinking that I could be righteous. But now seeing that with the Law sin abounds, for where knowledge of the Law is, sin increases; for the Law reveals our sin and man in his sin thinks he can obey God's law. So that I find in myself another law working in my members, the law of sin--the good that I want to do I don't do and the evil I don't want to do I do, and I find in myself only the wretchedness of my inability, my powerlessness, and my own rotting flesh. Here in my emptiness, my weakness, my nothing, as a wretched and filthy beggar, I find myself. And it is here as a beggar that God gives me all His gifts, in having nothing I have been given everything. So that, indeed, thanks be to God through Jesus Christ our Lord.
5. Can we be saved while continuing in known unrepentant sin?
Answer: We must be supremely careful here. For in one direction lay arrogance and in the other despair. Which is why a proper distinction and preaching of Law and Gospel is necessary. Yes, the Law reckons us sinners, and therefore as long as we go on in the destructive pursuits of our flesh we will surely die. And Scripture warns us repeatedly not to shipwreck our faith. However, we hear these words not to despair, to lose faith, and to suffer in hopelessness--but to be vigilent and always look to Christ. Because here in Christ is the indelible and irrevocable promise of God, that we belong to Him, that He keeps us, holds us, and makes us His own. And there is no power stronger than God's love which is ours in Christ, as St. Paul says, "I am convinced that neither life nor death, ... nor any other thing can separate us from the love of God which is in Christ Jesus" The call to repentance is not weapon to be wielded to manipulate people, or to be used to injure the faith of God's people, but instead is to be understood as our life of discipline, as cross-carrying disciples of Jesus. To keep our minds sober, to be full aware of our sins and our need for grace. So that we cleave to Jesus Christ and what He has done, and never seek to cling to our own works. For Christ alone saves, and He alone has saved us, and all which He has done is certain, true, and eternally satisfactory. Therefore we trust in Christ and not in ourselves.
And it is here that there is the new obedience. For our neighbor is hungry, let us feed him. Our neighbor is thirsty, let us give her drink. There are those in our midst who are suffering, who need medicine, who are being oppressed, who are lacking, who need clothes, and all manner of things. So the Lord calls us outward, outside of ourselves, and toward our neighbor. Christ does not call us to the self-righteousenss of the Judaizers who care about circumcision, sabbaths, and all such shadow. Christ calls us to the seriousness of the plight of our fellow man, created in God's image, who we are called to serve and minister in love even as Christ has loved us, even as Christ loves them. And here we do these things not out of hope for reward, but out of sacrifice and love. God will not reward you for feeding someone, you will get no shiny prize or a gold star on some heavenly record--you are to feed the hungry because they are hungry and it is the good thing to do. That and no other reason. Seeing as God "having forgiven us all our trespasses, by canceling the record of debt that stood against us with its legal demands" we should be free now, as slaves of God and not of sin, to love our neighbor. Knowing full well that the flesh remains and we, always, failing in our sin; but that nevertheless Christ saves us. We can therefore hope in Him, both now and on the last day.
-CryptoLutheran