Does the church claim to have any of Jesus’ actual teaching outside the bible?
If you are asking for the actual words of Jesus then no.
Jesus however, spoke to the Apostles for three years. The Apostles transmitted that teaching to their successors orally, and in some cases in writing. The teaching of the Apostles and their successors apart from the Bible is contained in Sacred Tradition.
Sacred Tradition contains many things revealed to us by God which are not in the Bible.
Yes, it’s included in Sacred Tradition. I don’t think there’s a list, though, that breaks down Tradition into “this came from Christ’s teaching, this came from his life, this from his institutions, this from the prompting of the Holy Spirit,” etc. So I don’t know if I can give you an example of exactly what you’re asking. Maybe someone else can.
The Catechism of the Catholic Church states:
76 In keeping with the Lord’s command, the Gospel was handed on in two ways:
orally “by the apostles who handed on, by the spoken word of their preaching, by the example they gave, by the institutions they established, what they themselves had received - whether from the lips of Christ, from his way of life and his works, or whether they had learned it at the prompting of the Holy Spirit”;33
in writing “by those apostles and other men associated with the apostles who, under the inspiration of the same Holy Spirit, committed the message of salvation to writing”.
77 "In order that the full and living Gospel might always be preserved in the Church the apostles left bishops as their successors. They gave them their own position of teaching authority."35 Indeed, "the apostolic preaching, which is expressed in a special way in the inspired books, was to be preserved in a continuous line of succession until the end of time."36
78 This living transmission, accomplished in the Holy Spirit, is called Tradition, since it is distinct from Sacred Scripture, though closely connected to it. Through Tradition, "the Church, in her doctrine, life and worship, perpetuates and transmits to every generation all that she herself is, all that she believes."37 "The sayings of the holy Fathers are a witness to the life-giving presence of this Tradition, showing how its riches are poured out in the practice and life of the Church, in her belief and her prayer."38
Thanks - that’s all I was looking for.
I appreciate the input.