Taking the last part of your essay first, you're misreading the sentence on unity. You've given it a meaning that isn't there.
"The present- day conditions of the world add greater urgency to this work of the Church so that all men, joined more closely today by various social, technical and cultural ties, might also attain fuller unity in Christ."
In no way can this sentence be interpreted as meaning that unity in Christ is simply a result of, or an extension of unity resulting from technical or cultural ties. On the contrary. It means the opposite. The sentence states clearly that these pre-existing cultural and technological ties make the work of the Church- what?? Yes, that's right-more urgent.
Surely urgency would be less of a necessity for the Church if that technological and cultural unity already existing naturally gave rise to unity in Christ? The writer is saying "no". The work of the Church is now more urgent than ever.
I see no indication whatsoever that there is a sequential connection between the two things in terms of one leading naturally to the other. Rather, I see the writer saying that the social human unity places even more demands on the Church and its spreading of its message.
As for the word "sacrament", the strict definition is "an outward sign of inner grace." However, my Catholic Encyclopedia also tells me:
Taking the word "sacrament" in its broadest sense, as the sign of something sacred and hidden (the Greek word is "mystery"), we can say that the whole world is a vast sacramental system, in that material things are unto men the signs of things spiritual and sacred, even of the Divinity. "The heavens show forth the glory of God, and the firmament declareth the work of his hands" (Ps. xviii, 2). The invisible things of him [i.e. God], from the creation of the world, are clearly seen, being understood by the things that are made; his eternal power also, and divinity" (Rom., i, 20).
One who defines "sacrament" only in its narrowest sense, will naturally take issue with the use of it in its broadest sense, which is what the Vatican document does-and let it be said rightly so. It is the preamble to a long document- preambles are when subject matter is introduced and terms and subject matter are generally defined in their broadest sense.
Finally, to claim that the Church is "a sign and instrument" of the "unity of the whole human race" does not deny Church teaching on any of the issues you list.
The unity spoken of in the passage is not a unity which results from all accepting Christ and obtaining salvation and none being damned. Rather it is a unity which results from the fact that:
Man is a creature of God in a created universe. All things that are, except Himself, exist in virtue of a unique creative act. Catholic Encyclpedia
The Church, being commissioned by Christ, to "make disciples of all nations", is indeed that sign and instrument.
Duh, I wonder why?
"Finally, to claim that the Church is "a sign and instrument" of the "unity of the whole human race" does not deny Church teaching on any of the issues you list."
Sure - and there is no evidence whatsoever that homosexuals in the priesthhod are a significant cause of the sex abuse scandals!!!
However, I believe your complacency is misplaced as even Paul VI felt the need to address this problem on 24th December 1965:
"The Church with its demanding and precise attitude to dogma, impedes free conversation and harmony among men; it is a principle of DIVISION in the world rather than of union. How are division, disagreement and dispute compatible with its catholicity and its sanctity?"
He tries to address this problem by saying that Catholicism is a principle of DISTINCTION among men rather than DIVISION. And he does this by reducing distinction to the level of that "natural unity" which I discussed above. He says that distinction is "of the same sort as that involved in the case of language, culture, art or profession." He goes on:
"It is true that Christianity can be a cause of separation and contrast, deriving from the good it bestows upon humanity: the light shines in the darkness and thus diversifies the zones of human space. But it is not of its nature to struggle against men, if it struggles at all, it is for them."
This is what I would call a tortured exegesis to try and reconcile two opposing worldviews.
Gone are the Scriptural principles of Christ being a "sign of contradiction", a "stumbling stone", a "sword that brings division" and warfare between the Church and the world.
So, if the Church is indeed an "instrument of unity" for all humanity, perhaps you can explain how this is so without negating the doctrines of predestination, hell, opposition between the Church and the world etc.?
Maybe you will make a better job of it than Paul VI did.