This is dang long, and in three parts. First, answering your points, Second, the problems that might crop up, and Third, the motivators for having children. All these motivators currently exist in our world, but will be more prevalent because children are a conscious choice rather than driven by sexual desire. This change will alter the fabric of your society.
You actually described sex when you said:
it's just that when a man and a woman decide they want to have a child together, they initiate the process and that makes the woman's body pregnant.
That there is the definition of sex. Two sets of genes and bio process.
What you are actually talking about is a lack of desire.
There's no rape, pedophilia, sexual harassment, etc.
You do realize that rape is not entirely driven by desire right? Likewise pedophila. If both must consent in order for reproduction to happen, then I do think it would cut down on it. But I can see a man or woman chaining someone in the basement and torturing them until they do consent so that they might have a kid. I can also see someone with a lot of money and status paying many people to have their babies.
There's no pornography, prostitution, etc.
No porn I can buy. But I can see women paying the most fit men for children. Which would be payment for sex and vice versa. You argue that no one pays anyone to have children in our world, therefore they would not in this one. But this world is radically different because of this change, that this is fairly likely to happen. (See my points below on children as status symbols).
Teenage pregnancy would be much less, although not zero since some teenagers might still decide that they want to have a child together.
Actually the pregnancy rate of teens might go up instead of down--if making a child is the most intimate thing you could do.
Birth control wouldn't exist, but abortion may still exist since a woman may decide in the middle of her pregnancy that she doesn't want a child
Despite the fact that there is no desire, the act itself might be deemed as an intimate exchange. So--might still be birthcontrol.
There would be no need to hide the truth of "where babies come from" from children, and thus there would be much less censorship in media.
If making a baby is easy, then there might still be censorship regarding the act itself. It might actually be a big secret! No one will imply or say how to do it because teens aren't ready for it--they only learn of it once they graduate. Plus, there is always SOMETHING people object to.
Large parts of major religions would be different, since the entire notion of sexual morality doesn't exist, but religions would still teach that you shouldn't have children out of wedlock
A notion of sexual morality can still exist in this world.
Relationships might be more honest, in the sense that someone won't pretend to love someone else just so that person will agree to have sex with them. But the phenomenon of gold-diggers, where someone pretends to love someone else to get their money, would still exist.
If you want the person so that they can give you babies only, doesn't need to be honest. Also, look at friendships, plenty of people AREN'T honest with their friends and use them for things despite no sexual attraction whatsoever.
Men may have less incentive to be successful in life, because they
don't have as much of a motive in impressing women in order to get
those woman to have sex with them. In fact I think Hugh Hefner argued
in "The Playboy Philosophy" that the entire reason civilization exists
is men doing great things in order to prove themselves to be worthy
mates to women.
The driver might actually be the chance to have offspring. If there is any advantage to that whatsoever (which there will be) there actually may be more requirements that males OR females set down in order to consider another for breeding. There HAS to be a drive of some sort, even if it isn't one of pleasure or lust, or else the species will die out in short order.
My main point is this: A world without sexual desire does not equal a utopia. All it really means is that you have new and different problems.
If you really do recall your childhood well, other than your needs being attended to by adults, it was NOT all roses and sunshine. There are a lot of interpersonal conflicts and cruelties that are meted out that have nothing to do with sexual desire.
Problems I foresee
MONEY, STATUS, AND THINGS MIGHT BE THE FOCUS
Marriage contracts will mainly have to do with money and providing for children. There may be less of a gender divide, so traditional gender roles may actually be totally destroyed. Instead there may be a drive to acquire resources and money. This drive does not have to biological in nature.
CHILDREN MAY BECOME STATUS SYMBOLS & A GOOD WAY TO RETIRE
What says plenty? The fact that you have a dozen kids running around. That's your legacy. And parents may actually demand a portion of their child's income over that child's lifetime, as a way to "pay back" the care they received as children. After all, the parent had a choice in the matter. They weren't accidents or born out of love. This practice of children providing for parents in retirement DOES happen in our world. It will be more frequent in this one because having a child is a choice and not borne from sex drive.
RELATIONSHIPS WILL NOT BE AS STRONG
You seem to believe that a lack of desire will make people more devoted to each other and likely to marry. Nothing can be further from the truth. Post-sex and during sex oxytocin is released bonding partners to each other (females more than males) and it's tied in with hormones and other things. Without desire, the business of creating children will often be just that--a business--something transactional rather than something borne out of real affection.
YOUR PEOPLE WILL NOT BE HUMAN
You can't just make humans, then subtract sexual desire and expect us to look remotely the same. Other instincts will come to the fore when it comes to behavior, and those things might not be so pleasant as you would like. You have build totally different sexual apparatus for them and those things might have problems of their own. You say in the comments that there will be gender differences, however, those differences that are seen in humans are actually driven by hormones in a VERY BIG WAY, so it's not realistic to have gender differences in the same way. Everything from bone density to musculature to secondary sexual characteristics, such as prominent breasts are influenced GREATLY by hormones. You will have to do a ton of research on this.
A HIGHLY MOTIVATED POPULATION
Ok, so evolution lesson here. You believe that males will get lazy without the motivation of sexual desire. But guess what? All those lazy fellows will not be passing on their genes. The ones who do will have a drive to make children, and their children and so on. And they will have to do whatever it takes to signal to the opposite sex that they can take care of children or they will not pass on their genes. When I say drive, I'm not necessarily talking biology, but your society, at large, will have to have motivation to create children and that motivation will be passed down, even if it's through societal expectations. They gotta want children. There have to be reasons.
A POSSIBLE LACK OF GENETIC DIVERSITY
If for some reason, your population only has a few in it that are motivated to have children, that means that only a few will. And if only a few do, their genes will get passed on, narrowing the gene pool so much that genetic defects will be rampant. This also makes them more prone to all die from disease, because a diverse population can withstand illness and viruses better. (Take, for instance the Black Death--1/3 of the population, those without resistance, died. Some got ill, then got better because they carried different genes). This problem will effect your society on a social level, because if genetic defects happen a lot, it will something they watch out for and plan around. They may even test for it before they have kids, and if they can't, they aren't together, if one of the two desires children and they don't want defects. Sorry, but biology has an impact on society.
REASONS TO HAVE KIDS
Because you want a friend.
This will result in a lot of lonely people desperate to have children. I can't imagine the dynamic in this case will be healthy, and I can, for sure, see a market for people paying to have kids, as outlined above. This happens in our world. The pain of childbirth is not a deterrent.
Because they will provide for you in old age.
In this society, as outlined above, this can be a reason to have them. If it isn't connected with sexual desire, said children should be grateful to you for their existence, because you CHOSE to have them. Not because of some desire that gave you pleasure, but because there's an advantage to it. This is actually a cultural reason people have had children and is still a tradition in China.
Because you want someone to continue the family business or
You only live so long, and because you brought junior into this world specifically for this, he darn well better follow in your footsteps. After all, you made him, because you chose it, and you may have done it specifically for this reason, unlike kids that are made currently, in our world. Even in our world, this does happen. It will happen a lot more in yours BECAUSE choice, not passion, is the motivator in your world. This will change society in that there may be lots of generational businesses. This is not prevalent as much today in America or the UK or Europe.
Because you want a permanent connection with another person,
using your offspring to do so.
Even without sexual desire, a desire to connect with another person and keep them in your life may be strong. In this case, you might lie or say whatever you need to in order to have that child, which would mean the person who had the child with you would be forever in your life. In our world, this does happen, and it isn't always tied to sexual desire.
Because you love the other person and think that you'll make a
wonderful child together
This seems to be the only reason you accept, but in your world, the reasons above will actually be more prevalent.
IN short--children will be MUCH more obligated to their parents. Your society will have much more respectful offspring because they all know that they were chosen to exist. While this is true when parents opt for fertility treatments in our world, it doesn't change the way society is run or how children are treated and seen because the majority of births are likely unscheduled/unplanned. If EVERYONE could chose to have kids and there was no drive, this would be the result.