In real life, spectators of judicial duels were forbidden to interfere by making any noise, and that included shouting, cheering and even coughing, under the penalty of death.
At the wedding, the clergy kiss each other, then one kisses Carrouges, then he kisses his bride. This was a real practice in a medieval Nuptial Solemn Mass; the "kiss of peace" was given by the priest celebrant to the deacon, who in turn gives it to the subdeacon, who gives the kiss to any other clergy present in choir dress, and it is also given to the groom, who then gives the kiss of peace to his bride. The kiss of peace was also given to royalty, if present at a Solemn Mass. At modern Catholic Masses, the sign of peace is usually a handshake, exchanged by all people present at the Mass.
Matt Damon struggled to grow facial hair, so the makeup department had to try out 20 different beards to apply to his face. He reportedly got a skin rash underneath the stuck-on beard, which was so irritating that he had to be treated with ice packs after every scene while shooting the movie.
Jean de Carrouges was born in the 1330s and Marguerite de Carrouges was born in 1362. They married in 1380. Thus Marguerite was 18 at her wedding and Jean was more or less 45. Also, Jean de Carrouges and Jacques le Gris were about the same age and were both squires and eventually knights, but at the time of the duel, Carrouges had participated in many more battles than le Gris, thus it can be surmised that he had better fighting reflexes and experience.
Scaffolding is visible around the cathedral of Notre-Dame de Paris. Although the cathedral was largely completed by 1270, over a century before the film is set, in the 14th century the flying buttresses that support the building's weight were being replaced with stronger versions.