First Question: Did the Second Vatican Council change the Catholic doctrine on the Church?
Response: The Second Vatican Council neither changed nor intended to change this doctrine, rather it developed, deepened and more fully explained it.
This was exactly what John XXIII said at the beginning of the Council. Paul VI affirmed it and commented in the act of promulgating the Constitution Lumen gentium: "There is no better comment to make than to say that this promulgation really changes nothing of the traditional doctrine. What Christ willed, we also will. What was, still is. What the Church has taught down
through the centuries, we also teach. In simple terms that which was assumed, is now explicit; that which was uncertain, is now clarified; that which was meditated upon, discussed and sometimes argued over, is now put together in one clear formulation". The Bishops repeatedly expressed and fulfilled this intention.
Second Question: What is the meaning of the affirmation that
the Church of Christ subsists in the Catholic Church?
Response: Christ "established here on earth" only one Church and instituted it as a "visible and spiritual community", that from its beginning and throughout the centuries has always existed and will always exist, and in which alone are found all the elements that Christ himself instituted. "This one Church of Christ, which we confess in the Creed as one, holy, catholic and apostolic […]. This Church, constituted and organised in this world as a society, subsists in the Catholic Church, governed by the successor of Peter and the Bishops in communion with him".
In number 8 of the Dogmatic Constitution Lumen gentium ‘subsistence’ means this perduring, historical continuity and the permanence of all the elements instituted by Christ in the Catholic Church, in which the Church of Christ is concretely found on this earth.
It is possible, according to Catholic doctrine, to affirm correctly that the Church of Christ is present and operative in the churches and ecclesial Communities not yet fully in communion with the Catholic Church, on account of the elements of sanctification and truth that are present in them. Nevertheless, the word "subsists" can only be attributed to the Catholic Church alone precisely because it refers to the mark of unity that we profess in the symbols of the faith (I believe... in the "one" Church); and this "one" Church subsists in the Catholic Church.
You can read more of the document here.