So that's it? If I passed this quiz, I can receive Communion?
Not exactly. The Church Code of Canon Law says that those who "have been excommunicated or interdicted after the imposition or declaration of the penalty and others obstinately persevering in manifest grave sin" are not to receive Communion and there are a LOT of grave (or mortal) sins. Catholics are supposed to examine their conscience and consult a priest.
Catechism of the Catholic Church 1857: "For a sin to be mortal, three conditions must together be met: "Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent."
What does the Church say will happen if I receive Communion anyway?
To respond to this invitation [to the Eucharist] we must prepare ourselves for so great and so holy a moment. St. Paul urges us to examine our conscience: "Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of profaning the body and blood of the Lord. Let a man examine himself, and so eat of the bread and drink of the cup. For any one who eats and drinks without discerning the body eats and drinks judgment upon himself." Anyone conscious of a grave sin must receive the sacrament of Reconciliation before coming to communion.
A person who is conscious of grave sin is not to celebrate Mass or receive the body of the Lord without previous sacramental confession unless there is a grave reason and there is no opportunity to confess; in this case the person is to remember the obligation to make an act of perfect contrition which includes the resolution of confessing as soon as possible.
TL;DR: It's a mortal sin, but only if you knowingly receive communion with the full understanding that you shouldn't be participating.
What about freedom of conscience? Aren't I free to make up my own mind about things like abortion and same-sex marriage?
Some Catholic commentators assert that a well-formed conscience and official Catholic teaching may come to opposite conclusions in moral matters. This opinion directly contradicts the Catechism, paragraph 2039: "Personal conscience and reason should not be set in opposition to the moral law or the Magisterium of the Church." A Catholic simply cannot claim to have a well-formed and well-informed conscience if he is ignorant of, misunderstands, or rejects outright God's law and thus commits acts that the Church considers gravely disordered.
Why are Catholics supposed to obey everything the Church says?
Why should we obey these teachings of the Church? Where does the Church get its authority? From Jesus. In Matthew 16:18-19, Jesus made Peter the head of his Church — the first Pope. He gave Peter and the Church "the keys to the kingdom of heaven":
"I say to you, you are Peter, and upon this rock I will build my church, and the gates of hell shall not prevail against it. I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven."
The Church's authority in faith and morals is absolute because Christ's authority is absolute.
Polls say that many American Catholics support same-sex marriage, contraception and other things on this list. Shouldn't someone stop them from receiving Communion?
Technically, the people themselves should refrain from receiving Communion if they're aware that their beliefs render them unable to receive the Eucharist in a state of grace.
Priests can only deny Communion to sinners whose sins are publicly known and politicians with professed beliefs contrary to Catholic teaching certainly fall into that category, as do notorious criminals or couples openly living together out of wedlock. St. Thomas Aquinas explains this in his Summa Theologica:
A distinction must be made among sinners: some are secret; others are notorious, either from evidence of the fact, as public usurers, or public robbers, or from being denounced as evil men by some ecclesiastical or civil tribunal. Therefore Holy Communion ought not to be given to open sinners when they ask for it... But if they be not open sinners, but occult, the Holy Communion should not be denied them if they ask for it. For since every Christian, from the fact that he is baptized, is admitted to the Lord's table, he may not be robbed of his right, except from some open cause.