Do public schools nurture democracy within their institutions? Are public schools part of democracy and does this occur in our schools?

Under ideal situations, yes, public schools do nurture democracy.  Students learn that their vote counts in student body elections.  Students often point out ambiguous questions on exams and, if valid, a teacher can modify the grade based on their complaints.  Many teachers give their students choices as to what the "fun" activity will be for the day.  By giving the student some autonomy over their day, the student feels more involved in the educational process.  Also, the public school classroom often serves as a good representation of American life. There are students of varying abilities and backgrounds in most classrooms.  
Day-to-day classroom management is usually not handled through a democratic process. The teacher makes most of the decisions.  Also, in some cases, the teacher does not let the student have any control over what they will do or the manner in which they will do it.  While this is not necessarily an incorrect way of management—all classrooms are different—many newer classroom management models are attempting to shift the learning model to being more "student-centered."