Just to warn you, you should limit your question so that it falls within the 20-year rule. We try to stay out of politics, and can't discuss anything that happened after 1996.

Christian opposition to abortion, homosexual (actually any non-penis-vagina) sex, and premarital sex has been fairly consistent since its early beginnings. Actually, abortion is a bit complicated: Christians have been essentially always opposed to abortion after "ensoulment" is said to occur. But Thomas Aquinas was also against abortion of unsouled fetuses, though he didn't regard it has homicide. Since the 1800s most Christians have believed that ensoulment takes place at conception. Generally though the official beliefs of traditional Christians haven't really changed much.

It's really that Western cultures have rejected many Christian theological, moral, and political teachings they used to accept in favor of philosophies rooted in the Enlightenment. (To see this, read Jeremy Bentham, an Enlightenment philosopher from the early 1800s. Most of his ideas are now central to Western culture and politics.) This process has involved social tension, revolutions, and wars. On the one hand, atheists, freethinkers, and gays (eg Oscar Wilde, Alan Turing) were persecuted by Christians in power. On the other hand, when the revolutionaries succeeded, they persecuted Christians (eg Vendee massacres during French Revolution). The issues you mentioned are areas where disagreement is the strongest, and so these issues are now emphasized by both sides.

In other words, we've witnessed a power struggle in the West. It's just that in France the struggle took place around the time of the French Revolution. In Germany the Kulturkampf (German for culture struggle) was fought in the latter half of the 19th century. In European countries with established Protestant churches (eg England), there wasn't much conflict, because the clergy actually were generally for the shift. But in America the culture war is not quite finished, due to the presence of the Catholic Church and non-mainline Protestant (aka evangelical) churches. (The mainline Protestant churches in America are sister churches of the establishment Protestant churches in Europe, and have supported the cultural shift from traditional Christianity towards Enlightenment values.) Also, unlike Europe, the American culture war has been waged more peacefully among "hearts and minds" especially via the media and in educational institutions.

If you follow progressive news sources (eg Huffington Post), you'll constantly be hearing about various offensive comments by conservative Christian Republican politicians or Catholic child sex abuse. You'll also hear about mostly secular young heroes from Teach for America or the Peace Corps. If you read evangelical or Catholic news sources (eg World Magazine or Crux Magazine), you'll constantly be reading stories about Christian charity work in America and abroad, the plight of refugees and immigrants, or celebratory profiles of gay Christians who attempt to be celibate. You'll also hear about child sex abuse in public schools (actually about as prevalent as that in Catholic church). In other words, there's a low-grade propaganda war going on in America.

I disagree with the premise of the question - that is to say, I think incorrectly attributes certain qualities upon a group that said group would reject.

Regardless of how one views these issues politically, I think we can all agree that American Christians do not perceive themselves to be anti-this or anti-that. The anti-abortion, anti-everything not penile-to-vaginal sexual intercourse between married adults, and other views are, from the perspective of believing Christians very much positive views. They would called it pro-life, with the life in question being that of the fetus, which Christians believe begins at conception. As for the sex stuff, it would be viewed as pro-morality, where public morality and virtue is a major component of public health more generally.

Regardless of whether we, personally and politically, believe that we might understand the true motives of the other side (e.g. many progressives would argue that the anti-abortion and anti-sex stuff is more about controlling women's bodies, even if those Christian voters deny this), we must, in a forum like AskHistorians, at least attempt to come at the question in a neutral way, and part of that, I would argue, means taking the claims of each sides at least at face value, even if we disagree with those claims or their implications.

As for the specifics of his question - why s/he never hears about the "nicer" parts and only about the "anti-" parts, I, like /u/aflexiblechain submit that OP is suffering from selection bias. OP is only getting his/her coverage of American Christianity through sources that are politically hostile to either American Christianity generally or the politics of the Christian Right specifically.

Might be better suited in /r/religiousdebates

I'll ask a related question that might answer OP's question. What are the origins of American Christianity having such a hold in American politics? Was it always like that?

Not sure if this helps, but this topic might answer your question. /u/dubstripsquads outlines how the roots of modern american fundamentalist Christianity.