"With" is the collocation for "the court" in this context. "With" has many nuances of meaning, but I would say that in this example, it's most like definition #2. http://learnersdictionary.com/definition/with "The court" is referred to as an entity, like a person. Another way of saying it is, "Pleadings are certain formal documents filed AT the court'S OFFICE that state the parties' basic positions." In this case, the "court's office" is a place rather than an entity.

This is a difficult question to answer. However, my sense is that "with" is part of legal language such as filing and indicates that the pleadings are taken possession of by the court and, therefore, "with" is the appropriate preposition rather than "at" or "in".