What was the justification of morality in Puritan Antinomianism? As far as I know, Puritans were devout Calvinists. According to Theopedia, Antinomianism is associated with disobedience to the established authority, so it may be connected with the idea of lawlessness, immorality, or licentiousness. Was this term merely a pejorative adjective for people who disagreed with the established religious authority, like legalism? Or did the Puritan antinomians really have their own theology on justification before God, and if so, how did the Puritan antinomians thought they were justified? What was the purpose of God's moral laws then?

This question was previously posted on the History.SE, but it was closed for being off-topic. Hopefully, it's on-topic here.
From the theopedia article you refer to:

Antinomianism ... is a pejorative term for the teaching that Christians are under no obligation to obey the laws of ethics or morality. Few, if any, would explicitly call themselves &quot;antinomian,&quot; hence, it is usually a charge leveled by one group against an opposing group. (emphasis added)

and:

Calvinists have also drawn charges of antinomianism. In the history of American Puritanism, Roger Williams was accused of antinomian teachings by the Puritan leadership of New England.
Theological charges of antinomianism typically imply that the opponent's doctrine leads to various sorts of licentiousness, and imply that the antinomian chooses his theology in order to further a career of dissipation. The conspicuous austerity of life among surviving groups of Anabaptists or Calvinists suggests that these accusations are mostly for rhetorical effect. (emphasis added)

In the particular example cited of Roger Williams, he was a separatist and had a different conception of religious liberty than many of his contempories, however, an unqualified charge of 'antinomianism' against him is not well supported: His efforts to secure a charter for the colony on Rhode Island and to serve as it's governor there demonstrate clearly that he did believe in civil law and civil authorities.
In summary, the answer to:

Was this term merely a pejorative adjective for people who disagreed with the established religious authority, like legalism?

Is Yes.
This renders your other questions somewhat moot, as well as being based on either a false or confused premise:  Protestants in general, including the Puritans, do not believe in justification by the law. They believe that the law was given to restrain wickedness, convict of sin and lead men to a knowledge of their need for a saviour. Disagreements about the right use of the law stem more from arguments over whether it has an ongoing utility in the spiritual life of a believer (eg in regards to gaining assurance of salvation, as a guide during the process of sanctification etc.) rather than whether it has any role in society at large and it's application towards Christians as citizens.