The church of Christ was founded by Christ. The doctrine of the church is based on the teachings of Christ and His Apostles. In the sense that "catholic" means universal, I don't disagree. There is one church, founded by Christ, and it's only head is Christ. That is the church that God added me to on the day I obeyed the Gospel.
That is correct and Peter is the rock on which the Church was built by Christ. There is only one church that traces its ancestry back to Christ and that is the Catholic Church.