Short answers, pulled from the story.
A dispensation is the exemption from the immediate obligation of the law in certain cases. Its object is to modify the hardship often caused by rigorous application of general laws to particular cases, and its essence is to preserve the law by suspending its operation in such cases.
The actual practice of the Roman Catholic Church is based upon the decisions of the Council of Trent which left the medieval theory intact while endeavouring to guard against its abuses. The canons of the council of Trent were decreed saving the authority of the Holy See in session twenty-five chapter twenty-one de ref.
The pope exercises his power of dispensation through the Roman Congregations and Tribunals as the Vicar of Christ with divine authority and jurisdiction. Bishops can dispense from impediments not reserved to the pope but diocesan bishops may dispense in very urgent danger of death from all diriment impediments except priesthood and affinity from lawful intercourse in the direct line.
Dispensations should be free of all charges according to the Council of Trent decree in session twenty-four chapter five De ref matrim. Expenses incurred by petitioners fall under four categories including carriage expenses, a tax to defray administration costs, a fine called componendum applied to pious uses, and an alms imposed on petitioners.
Payment of money for a dispensation would void the dispensation as laid down in respect of all dispensations whether papal or other that they were to be granted only for just and urgent causes or in view of some decided benefit to the Church and in all cases for free.