(A) If a proceeding for divorce, dissolution, legal separation, annulment, or the allocation of parental rights and responsibilities for the care of a child involves one or more children, if the parents of the children do not agree upon an appropriate allocation of parental rights and responsibilities for the care of their children or do not agree upon a specific schedule of parenting time for their children, the court may order the parents to mediate their differences on those matters in accordance with mediation procedures adopted by the court by local rule. When the court determines whether mediation is appropriate in any proceeding, it shall consider whether either parent previously has been convicted of or pleaded guilty to a violation of section 2919.25 of the Revised Code involving a victim who at the time of the commission of the offense was a member of the family or household that is the subject of the proceeding, whether either parent previously has been convicted of or pleaded guilty to an offense involving a victim who at the time of the commission of the offense was a member of the family or household that is the subject of the proceeding and caused physical harm to the victim in the commission of the offense, and whether either parent has been determined to be the perpetrator of the abusive act that is the basis of an adjudication that a child is an abused child. If either parent has been convicted of or pleaded guilty to a violation of section 2919.25 of the Revised Code involving a victim who at the time of the commission of the offense was a member of the family or household that is the subject of the proceeding, has been convicted of or pleaded guilty to any other offense involving a victim who at the time of the commission of the offense was a member of the family or household that is the subject of the proceeding and caused physical harm to the victim in the commission of the offense, or has been determined to be the perpetrator of the abusive act that is the basis of an adjudication that a child is an abused child, the court may order mediation only if the court determines that it is in the best interests of the parties to order mediation and makes specific written findings of fact to support its determination. If a court issues an order pursuant to this division requiring mediation, it also may order the parents to file a mediation report within a specified period of time and order the parents to pay the cost of mediation, unless either or both of the parents file a motion requesting that the court waive that requirement. Upon the filing of a motion requesting the waiver of that requirement, the court, for good cause shown, may waive the requirement that either or both parents pay the cost of mediation or may require one of the parents to pay the entire cost of mediation. Any mediation procedures adopted by local court rule for use under this division shall include, but are not limited to, provisions establishing qualifications for mediators who may be employed or used and provisions establishing standards for the conduct of the mediation.