The role of monks and their appellation in 18th century?
Technically speaking Dominicans are clerical canons and are called friars and not monks. They live in convents and not monasteries. Typical Dominican priests generally remain in their respective convents for six months of the year and preach the faith to the people for the other six months of the year. Nowadays variations to these time frames varies according to various situations.
The Order of Preachers, also known as the Dominicans, is an order of the Catholic Church founded in Toulouse, France, by the Spanish priest Dominic of Caleruega. It was approved by Pope Honorius III via the papal bull Religiosam vitam on 22 December 1216. Members of the order, who are referred to as Dominicans, generally carry the letters OP after their names, standing for Ordinis Praedicatorum, meaning of the Order of Preachers. Membership in the order includes friars, nuns, active sisters, and affiliated lay or secular Dominicans (formerly known as tertiaries, though recently there has been a growing number of associates who are unrelated to the tertiaries).
Founded to preach the Gospel and to oppose heresy, the teaching activity of the order and its scholastic organisation placed the Preachers in the forefront of the intellectual life of the Middle Ages. The order is famed for its intellectual tradition, having produced many leading theologians and philosophers.
In England and other countries, the Dominican friars are referred to as Black Friars because of the black cappa or cloak they wear over their white habits. Dominicans were "Blackfriars", as opposed to "Whitefriars" (i.e., Carmelites) or "Greyfriars" (i.e., Franciscans). They are also distinct from the "Austin friars" (i.e., Augustinian Friars) who wear a similar habit.
Saint Dominic (1170–1221), portrait by El Greco, about 1600
A priest of the Dominican is typically addressed as ”Father”. The Lay Brothers normally remained in the convents to maintain to the needs of the community: cooking, sewing, etc. Although dressed in a all white habit, the put on a black cape while outside of the convent. Dominicans are not choir monks in the least.
And yes, choir monks, such as Benedictine monks, may leave their monasteries and are usually addressed as ”Father” also.
The image in your question is historically inaccurate in making a Dominican dressed in a red habit. This is pure fiction.
"The cruel death of Calas, who was broke on the wheel at Toulouse, 10 March 1762."
Jean Calas (1698 – 10 March 1762) was a merchant living in Toulouse, France, who was tried, tortured and executed for the murder of his son, despite his protestations of innocence. Calas was a Protestant in an officially Catholic society. Doubts about his guilt were raised by opponents of the Catholic Church and he was exonerated in 1764. In France, he became a symbolic victim of religious intolerance, along with François-Jean de la Barre and Pierre-Paul Sirven.
It should be noted that his trial and execution was done by the state authorities and the Church did not take part in the condemnation of Jean Calas. Unfortunately it occurred in a Catholic State and when tolerance of other religions was not a true it’s best.
The Catholic Encyclopedia has this to say about his condemnation:
The Calas Case was not without its effect on contemporary art and literature. Over a hundred publications relating to it are in existence. It forms the subject of many plays by F.-L. Laya (produced for the first time in Paris in 1790), Lemierre d'Argy (Paris, 1790), Marie-Joseph Chénier (Paris, 1791), and Victor Du Cange (Paris, 1819). Madame Calas and her daughters were living in Paris, when several of these were presented on the stage. Some historians, carried away perhaps by too great a desire to bring the innocence of Jean Calas to the fore, assert that Marc-Antoine committed suicide. But there are weighty reasons to doubt the father's innocence (Barthélemy). Voltaire cannot be considered an impartial historian of the case, owing to his preconceived desire to present a strong indictment against the Catholic Church, rather than to state the facts in their true light. The responsibility of the condemnation in no way rested with the ecclesiastical authorities, and the penalty was inflicted not for a mere religious offence, but for murder alleged to have been committed for a religious motive.