Why does Shakespeare make Lord Capulet a dynamic character?It is supposed to have to do with the church splitting (protestant and catholic) and the other history of his time.
To answer the first part of the question, Capulet as well as Montague can both be seen as dynamic characters because they begin the play as enemies, though what they are feuding over is never revealed, and end the play on peaceful terms, both changing their outlook on their feud as they recognize that Romeo and Juliet are "poor sacrifices to [their] enmity."
The most obvious reason as to why Shakespeare chose to make them dynamic is that he had to have them recognize their mistake. Shakespeare was probably familiar with Aristotle's Poetics in which Aristotle discusses the elements of a good tragedy. Recognition is one of those elements. The hero has to recognize his mistake (known as hamartia) and thus understand the irony of his situation and why his catastrophe occurred. Shakespeare does not strictly adhere to Aristotle's rules, but applies them according to his needs. In the case of Romeo and Juliet, the two heroes are dead and unable to have that moment of recognition. It isn't until Friar Lawrence reveals the story of Romeo and Juliet to Capulet and Montague that they see the cost of their feud and make amends, although too late.
Another reason to make them dynamic is that had they not changed and recognized the futility and cost of their feud, then the deaths of their children would have been for nothing, and the denouement of the play would be unsatisfactory. As it is written, Friar Lawrence's desire to see the feud end through the union of Romeo and Juliet is achieved, but obviously not in the manner he imagined. Nevertheless, the audience can take some satisfaction in knowing that the feud between the Montagues and the Capulets is resolved.
As to the second part of the question, it is known that Shakespeare's family was Catholic and that Catholics were routinely persecuted during Elizabethan times. In addition, Shakespeare did include some implicit political and social commentary in his work, especially the history plays. However, there are a few reasons why this may not be the case in Romeo and Juliet. First, the story itself first appeared in Italy before Shakespeare's time and the Protestant Reformation. He most likely adapted the story from Arthur Brooke's poem, The Tragicall Historye of Romeus and Juiet, one of the many versions that were in circulation since its original publication in 1476 under the title Il Novellino by Masuccio Salernitano ("A Note on the Source of Romeo and Juliet," Signet Classics). Second, in the play, both families are connected to Friar Lawrence who is presumably a member of the Catholic clergy given that the play is set in Verona, Italy, a country that at the time was and still is predominately Catholic.