Luke, the answers to your questions are simple enough, but those answers do not bring solutions to the problems.
1. Why are so many young people not proactive about contraception when information is so readily available beyond just public schools and private homes?
First, young people are stupid. Maybe you were not; you might have been an old man when you were 13. In criminology, we say that most offenders age out of crime. Family, work, life, tend to mitigate against the "idle hands" factor. It is sort of an externality to "infant mortality": if you survive childhood, you live longer -- tautologously enough... Shoplifting, drug use and abuse, reckless driving, gangs, truancy, sex, weird clothing, disrespect for authority, you know, it is all pretty much a package.
2. What would it be like if all education were private and sex education became a private matter, e.g. would we see a rise or drop in unplanned pregnancies and related poverty?
Did Romeo and Juliette make good choices? Education was private for millennia. Sex education was nearly non-existent.
Most people understood the relationship between coitus and pregnancy in large terms. But not every act resulted in a pregnancy. Phases of the moon, herbs, and actual ignorance were large factors in our understanding.
While the non-mammalian animal egg was obvious, the doctrine ex ova omne vivum ("every living [animal comes from] an egg"), associated with William Harvey (1578-1657), was a rejection of spontaneous generation and preformationism as well as a bold assumption that mammals also reproduced via eggs. Karl Ernst von Baer discovered the mammalian ovum in 1827, and Edgar Allen discovered the human ovum in 1928. The fusion of spermatozoa with ova (of a starfish) was observed by Oskar Hertwig in 1876. -- https://en.wikipedia.org/wiki/Egg_cell
Those were people who lived close to farm animals. Growing up in the city, I assure you, as interesting as girls were, I had no perceptual knowledge of sexual intercourse. I will grant you, though, that as soon as I got the lecture, the rest was clear, and I managed to avoid an unwanted pregnancy (terminated) until I was 25. ("What do you mean, you're not on the pill?") And that underscores something about the statistics. Despite "alarming numbers" we are still talking about minorities of people. "That means that 34 percent of teenagers have at least one pregnancy before they turn 20." -- http://www.teenhelp.com/teen-pregnancy/teen-pregnancy-statistics.html From the same source, 15% to 25% of teen pregnancies result in miscarriage. Also, note that one-third of pregnant teens finish high school. So, we have to moderate our sense of panic.
When I was in the second grade (1957), we experienced a radical change in school board policy when our teacher continued to teach while pregnant. That reflected the attitudes about (ahem) "health classes" in school. And we had teen pregnancies, of course. People have had children out of wedlock since the invention of wedlock.
Finally, my brother and his wife had to wait for her to turn 16 so that they could get a court order to get married. They were married for 30 years before she died. I could go on and on about their lifecourse successes. But that speaks to your third point.
3. If the average young person really is this stupid and needs this much state coddling, where does this leave Objectivism as a philosophy that sees the individual as a sovereign entity from a young age forward?
They were both Objectivists, more or less. Paul read Ayn Rand shortly after I did. Jill took it up about a decade later and borrowed a bunch of materials from me from the early days. It is not that Ayn Rand saved their lives, but that they were thinking people, even if they had been impulsive teenagers.
Objectivism does not grant legal carte blanche to every person who asserts free will. Even in Galt's Gulch, it might be socially disgraceful - if not illegal - to sell tobacco to a minor. I think that everyone who pays taxes should vote. So did Herbert Spencer, and he included children in that: children who work for wages should vote -- but not necessarily drive a car or join the army. You cannot be President until you are 35. Some of them were immature nonetheless, judging from their sexual escapades.