In the example of the murderer, would God expect him to turn himself in to the authorities if his crime was not known or found out? 

One aspect of repenting of sin is demonstrating a complete change in life. "For godly sorrow produces repentance leading to salvation, not to be regretted; but the sorrow of the world produces death. For observe this very thing, that you sorrowed in a godly manner: What diligence it produced in you, what clearing of yourselves, what indignation, what fear, what vehement desire, what zeal, what vindication! In all things you proved yourselves to be clear in this matter" (II Corinthians 7:10-11).

What matters most is clearing ourselves with God, yet there are many sins which are also crimes against man. While we should rectify the damage our sins may have caused, when possible, the reality is that in many cases the damage cannot be undone or even lessen.

When the Jews brought to Jesus a woman caught in the act of adultery, there was no question that the woman was involved in sin (John 8:3). Both the ones who brought her to Jesus and even Jesus, himself, stated she had sinned. The problem was that Jews who were accusing her had broken the law in two ways. One, they had only brought the woman. Though caught in the act of adultery, they did not bring the man. Second, the law required that they bring such people before the courts and they brought her to Jesus. When Jesus forced them to face their sins by what he stated, they left. "Then those who heard it, being convicted by their conscience, went out one by one, beginning with the oldest even to the last. And Jesus was left alone, and the woman standing in the midst" (John 8:9).

When Jesus and the woman were alone, Jesus asked, "Woman, where are those accusers of yours? Has no one condemned you?" (John 8:10). Though Jesus had the right to judge the woman, and though He knew what she had done, he followed the law. The law required that a crime be witnessed by two or more. In addition, the judge cannot be one of the witnesses, nor does the law require a person to be witness against himself. Therefore, no case can be made when there were no witnesses. Jesus told her, "Neither do I condemn you; go and sin no more" (John 8:11).

The woman had committed a crime that was punishable in the courts by man, but because it could not tried by men, Jesus focused on the important part -- stop sinning.

I cannot find a command that sinners whose sin also broke a civil law is required to turn themselves in. They should be aware that God will punish sinners who do not leave their sins behind. Sins don't often stay hidden. "Evil pursues sinners, but to the righteous, good shall be repaid" (Proverbs 13:21). And nothing is hidden from God. "You have set our iniquities before You, our secret sins in the light of Your countenance" (Psalms 90:8). A person needs to repent of their sins and follow God's will. If as a result of their prior sins there is later an accounting, then it must be faced at that time. But it is better to face a court of men than to face God in judgment with unwashed blood on your hands.

See also:

Questions and Answers regarding Sin