Most complaints need not be verified. A case in Family court is called a proceeding. The parties to the action are the petitioner and the respondent. The plaintiff's pleading is a verified petition. A trial in Family Court is called a hearing.
Many proceedings have two hearings, a fact finding hearing, and a dispostional hearing.
Different rules of evidence applies these hearings. There is no right to have assigned counsel in a Supreme Court case. For matrimonial and support cases, it is possible to have the court make an award of counsel fees which requires the spouse with more money or assets pay the counsel fees for the other spouse. The Family Court also has the authority in some instances to award counsel fees.