To your first question, I have never heard “Catholic” associated with forced compliance via contract.
If I focus less on the 'school" part of the description and more on the “Catholic” part of the description, as you suggest, the slippery slope becomes even more severe. Would you favor the same sort of contracts for all employees at a Catholic hospital? I certainly would not. I find the entire notion of this contract very unseemly.
Teachers are role models for their students, who are still learning about the world and what life in it is like. Catholic hospitals have a different mission, and patients are not shaped by its staff as happens in schools.
In the secular world, firing an an employee for off-hours behavior is not uncommon. Even someone in a modest position, such as a bank teller, have been terminated because her (non-criminal) behavior in another state made the news.
In the case of teachers, the archdiocese is not requiring that the employee be Catholic. However, the teacher is expected to advance the mission of the schools. The new contract makes this more explicit than past contracts. However, the mission of Catholic schools has always been to shape students so they develop as Christians and are able to carry their faith into adult life.
This mission is not unique to Cincinnati. If you check out the mission statement of any Catholic school system, I think you will find that to be true. Here are a few, taken at random:
Archdiocese of New York
The mission of the Catholic Schools of the Archdiocese of New York is to ensure our schools are Christ-centered, academically excellent, and welcoming communities that teach students to be life-long learners and leaders energized by fidelity to Christ, the Church, and one another
adnyeducation.org/about-us/mission-statement/
Archdiocese of San Francisco
*The Department of Catholic Schools of the Archdiocese of San Francisco is dedicated to the teaching mission of the Catholic Church. For our elementary and secondary schools we strive to provide quality leadership, to develop programs of educational excellence, and to prepare students for a truly Christian life. *
sfdcs.org/dcs/about
Archdiocese of Galveston-Houston
*Catholic schools call young people to holiness and prepare them to live and proclaim the Gospel of Jesus Christ.
In collaboration with families, Catholic schools teach the values of our faith and tradition, foster experiences of community and service, and provide educational excellence. *
archgh.org/cso/
The “faith and morals” clause has always existed in the teacher contracts of Catholic schools. It is one of the things which sets Catholic schools apart from other schools. The Archdiocese of Cincinnati is simply tightening up the language of the contract so it will stand up in court.