With reference to the movie The Grand Budapest Hotel, what​ ​is​ ​the​ ​role​ ​films​ ​play​ ​in​ ​perpetuating​ ​stereotypes​ ​in​ ​contemporary​ ​society?

There are a number of characters in The Grand Budapest Hotel who conform to stereotypes, at least to some extent. Two of the most obvious examples are Monsieur Gustav, the protagonist; and Agatha, the female romantic interest of Zero, the narrator.
The gay or bisexual concierge, Monsieur Gustav, is to some extent a stereotypical representation of a gay or bisexual man. He is appalled by the wrong shade of nail varnish, is described as always "liberally perfumed," possesses "a library of romantic poetry," in his own words was "once considered a great beauty," and in the words of Dimitri is a "goddamn little fruit." In other words, Monsieur Gustav lives up to the stereotype of a gay man as a well-groomed, sensitive aesthete. It's worth noting, however, that he is also a very charming, intelligent, and kind character. He is not in the film, as so many gay characters often are, as merely a comic turn or figure of fun.
Agatha is a young, beautiful pastry chef, introduced ostensibly as Zero's romantic interest. Stereotypically, the female romantic interest is important only or primarily in as much as she contributes to the characterization of the male lead. Such a character is usually dependent upon the male lead and doesn't have many, if any, scenes on her own. Agatha, to some extent, fulfills this stereotypical role. She only appears in a few scenes, and in all of them she is with Zero or Gustav. The first time we meet Agatha she is in a kitchen, and one of the last times we see her is when she is getting married to Zero. This seems to comply with the stereotypical representation of women as belonging to the domestic sphere, in the kitchen and as wives. Agatha does, however, in some ways defy the stereotype of the female romantic interest. She is brave and courageous, helping Gustav, for example, to smuggle tools into prison, and she also speaks her mind.
One good method of testing to see how far women are represented stereotypically in film is to apply the Bechdel test. This test poses a number of questions. For example, do the female characters talk to each other? And if so, do they talk about something other than men? It's interesting how many films, even in 2019, don't pass this simple test. Women in film are still very often presented as little more than auxiliaries to the male characters, perpetuating the stereotype that women are not as important, active, or independent as men.
In general, films play a huge role in perpetuating all kinds of stereotypes in films. Marlon Brando, one of the most famous actors of all time, was a vocal critic of, for example, Hollywood's representation of minorities. In 1973, Brando boycotted the Academy Awards and sent Sacheen Littlefeather, a Native American woman, in his place. He did this to protest and raise awareness about the representations of minorities, specifically Native Americans, in Hollywood films.