Are the protagonist and antagonist of the Gospel of Matthew dynamic or static, round or flat characters? How do these characters change or remain...

First, one should note the Gospel according to Matthew is not a literary work, but instead a religious text in the form of a biography. This means it doesn't really follow the conventions of traditional literary works and many of the terms of literary criticism are inappropriate to it. Instead, Biblical scholars tend to prefer rhetorical analysis, as biography, in the period in which this work was composed, was considered part of epideictic rhetoric. 

The...

First, one should note the Gospel according to Matthew is not a literary work, but instead a religious text in the form of a biography. This means it doesn't really follow the conventions of traditional literary works and many of the terms of literary criticism are inappropriate to it. Instead, Biblical scholars tend to prefer rhetorical analysis, as biography, in the period in which this work was composed, was considered part of epideictic rhetoric. 

The protagonist of the work is Jesus. Although he appears as a baby in the beginning of the work and an adult at the end, he is not a traditionally dynamic character, as he is treated as the incarnate logos, who exists eternally, takes human form, and then returns to his heavenly origins. Although as the Son of Man, he does exhibit human form and occasional elements of humanity, his divine nature makes him neither a fully rounded nor flat character in a conventional literary sense, but rather something different than either because he has a divine or metaphysical interiority rather than a human psychology. His choices are ordained in advance by various prophecies, and as such are not really choices; they are often accounted for simply as necessary to fulfill prophecies. Since Jesus has complete and perfect foreknowledge of all of the events in the text, one can't really think of his being affected by experiences in the way a purely human character would be. 

The series of antagonists Jesus encounters, including Herod and Satan, are mainly flat characters, emblems of pure evil. Pilate is an interesting character, neither fully good nor evil, but rather an overworked bureaucrat. Pilate appears too briefly to be considered fully rounded, though. Peter is among the most fully rounded of the disciples in the way he struggles with doubt and pride.