b. Isn't this Old Testament law?

Some may argue that this law is contained in the Old Testament and that the New Testament did away with the law. However, the death penalty instituted by God in Genesis 9:6 was before Mosaic law and was not meant for one dispensation but "for all generations to come" (Gen 9:12). Jesus said that He did not come to "abolish the law but to fulfill it" (Matt 5:17). The divine authority invested in the government is repeated in the New Testament where it says, in Romans 13, that the civil government "does not bear the sword for nothing. He is God's servant, an agent of wrath to bring punishment on the wrongdoer ... to punish those who do wrong". Keep in mind that the sword is a weapon (and this was well understood in the Roman times) which is capable of inflicting all sorts of injury, including death.

In the story of the adulterous woman in John 8, Jesus did not say that the woman should not be executed for her crime, but rather, he challenged the morality of her accusers, i.e., if they were also guilty of the same crime, they could not lawfully put her to death without also bringing the same penalty upon themselves.

Israel, as a captive nation, was not living under Old Testament law. The right of "high justice" belonged only to the Roman government. Thus, by seeking to have Jesus commend the crowd's death penalty, the religious leaders were seeking to put Him in conflict with Rome as a revolutionary. On the other hand, they hoped to catch Him with the other extreme by making Him deny the Old Testament. Jesus saw the trap and resisted both extremes. Finally, the Old Testament requires two witnesses for a capital crime. Thus, when her accusers slunk away in shame (possibly because some of them had also committed adultery with the same woman) there were no witnesses, and therefore no case.

It is because God is a God of love that He provides the means to protect society form public evil. It is when the Divine commandment to "love your neighbour" breaks down in the heart of man, that the civil government's sword of the death penalty is designed to uphold and protect the sacredness of life.

The Lord is love personified but He is also a God of justice. Under His justice we are all sentenced to death for our personal sins. His love, which forgives the repentant sinner, is perfectly in harmony with His justice which insisted on capital punishment for His beloved Son, Who paid the penalty for sin by bearing the ultimate punishment - death and separation from God.
http://www.acdp.org.za/index.php?page=issue4