Here is the Pope on the use of the Shepherd's rod:
The Church too must use the shepherd’s rod, the rod with which he protects the faith against those who falsify it, against currents which lead the flock astray.
The Catechism explains why ordination is only conferred on men and not women.
1577 "Only a baptized man (vir) validly receives sacred ordination." The Lord Jesus chose men (viri) to form the college of the twelve apostles, and the apostles did the same when they chose collaborators to succeed them in their ministry. The college of bishops, with whom the priests are united in the priesthood, makes the college of the twelve an ever-present and ever-active reality until Christ's return. The Church recognizes herself to be bound by this choice made by the Lord himself. For this reason the ordination of women is not possible.
1578 No one has a right to receive the sacrament of Holy Orders. Indeed no one claims this office for himself; he is called to it by God. Anyone who thinks he recognizes the signs of God's call to the ordained ministry must humbly submit his desire to the authority of the Church, who has the responsibility and right to call someone to receive orders. Like every grace this sacrament can be received only as an unmerited gift.
1579 All the ordained ministers of the Latin Church, with the exception of permanent deacons, are normally chosen from among men of faith who live a celibate life and who intend to remain celibate "for the sake of the kingdom of heaven." Called to consecrate themselves with undivided heart to the Lord and to "the affairs of the Lord," they give themselves entirely to God and to men. Celibacy is a sign of this new life to the service of which the Church's minister is consecrated; accepted with a joyous heart celibacy radiantly proclaims the Reign of God.
This is the Church's teaching on abortion:
2319 Every human life, from the moment of conception until death, is sacred because the human person has been willed for its own sake in the image and likeness of the living and holy God.
2322 From its conception, the child has the right to life. Direct abortion, that is, abortion willed as an end or as a means, is a "criminal" practice (GS 27 § 3), gravely contrary to the moral law. The Church imposes the canonical penalty of excommunication for this crime against human life.
2270 Human life must be respected and protected absolutely from the moment of conception. From the first moment of his existence, a human being must be recognized as having the rights of a person - among which is the inviolable right of every innocent being to life.
- Before I formed you in the womb I knew you, and before you were born I consecrated you. My frame was not hidden from you, when I was being made in secret, intricately wrought in the depths of the earth.
2271 Since the first century the Church has affirmed the moral evil of every procured abortion. This teaching has not changed and remains unchangeable. Direct abortion, that is to say, abortion willed either as an end or a means, is gravely contrary to the moral law:
- You shall not kill the embryo by abortion and shall not cause the newborn to perish. God, the Lord of life, has entrusted to men the noble mission of safeguarding life, and men must carry it out in a manner worthy of themselves. Life must be protected with the utmost care from the moment of conception: abortion and infanticide are abominable crimes.
2273 The inalienable right to life of every innocent human individual is a constitutive element of a civil society and its legislation:
"The inalienable rights of the person must be recognized and respected by civil society and the political authority. These human rights depend neither on single individuals nor on parents; nor do they represent a concession made by society and the state; they belong to human nature and are inherent in the person by virtue of the creative act from which the person took his origin. Among such fundamental rights one should mention in this regard every human being's right to life and physical integrity from the moment of conception until death."
"The moment a positive law deprives a category of human beings of the protection which civil legislation ought to accord them, the state is denying the equality of all before the law. When the state does not place its power at the service of the rights of each citizen, and in particular of the more vulnerable, the very foundations of a state based on law are undermined. . . . As a consequence of the respect and protection which must be ensured for the unborn child from the moment of conception, the law must provide appropriate penal sanctions for every deliberate violation of the child's rights."
2274 Since it must be treated from conception as a person, the embryo must be defended in its integrity, cared for, and healed, as far as possible, like any other human being.
Prenatal diagnosis is morally licit, "if it respects the life and integrity of the embryo and the human fetus and is directed toward its safe guarding or healing as an individual. . . . It is gravely opposed to the moral law when this is done with the thought of possibly inducing an abortion, depending upon the results: a diagnosis must not be the equivalent of a death sentence."
Here is the Catholic Church's position on both homosexuality and marriage.
2357 Homosexuality refers to relations between men or between women who experience an exclusive or predominant sexual attraction toward persons of the same sex. It has taken a great variety of forms through the centuries and in different cultures. Its psychological genesis remains largely unexplained. Basing itself on Sacred Scripture, which presents homosexual acts as acts of grave depravity, tradition has always declared that "homosexual acts are intrinsically disordered." They are contrary to the natural law. They close the sexual act to the gift of life. They do not proceed from a genuine affective and sexual complementarity. Under no circumstances can they be approved.
2358 The number of men and women who have deep-seated homosexual tendencies is not negligible. This inclination, which is objectively disordered, constitutes for most of them a trial. They must be accepted with respect, compassion, and sensitivity. Every sign of unjust discrimination in their regard should be avoided. These persons are called to fulfill God's will in their lives and, if they are Christians, to unite to the sacrifice of the Lord's Cross the difficulties they may encounter from their condition.
2359 Homosexual persons are called to chastity. By the virtues of self-mastery that teach them inner freedom, at times by the support of disinterested friendship, by prayer and sacramental grace, they can and should gradually and resolutely approach Christian perfection.
2360 Sexuality is ordered to the conjugal love of man and woman. In marriage the physical intimacy of the spouses becomes a sign and pledge of spiritual communion. Marriage bonds between baptized persons are sanctified by the sacrament.
892 Divine assistance is also given to the successors of the apostles, teaching in communion with the successor of Peter, and, in a particular way, to the bishop of Rome, pastor of the whole Church, when, without arriving at an infallible definition and without pronouncing in a "definitive manner," they propose in the exercise of the ordinary Magisterium a teaching that leads to better understanding of Revelation in matters of faith and morals. To this ordinary teaching the faithful "are to adhere to it with religious assent" which, though distinct from the assent of faith, is nonetheless an extension of it.
The Church's teaching on abortion,contraception, homosexuals, homosexual "marriage", and women's ordination are all infallible teachings of the Catholic Church.