Chris’ answer as far as recent changes in pronoun usage goes is not accurate for some parts of the US. Here, there is a growing, but controversial, trend of all people identifying the pronoun by which they want to referred. This is done to be inclusive so that people who don’t identify as either male or female or whose gender isn’t obvious aren’t referred to by the wrong pronoun and aren’t singled out in having to identify their preferred gender. For example: ‘My name’s David and I use the pronouns ‘he’ and ‘him’. ‘My name is Edd, I go by they/them.’ This has nothing to do with the long-standing singular ‘they’ which is used in situations where gender doesn’t matter. It isn’t perceived as offensive. ‘A lawyer called and they want you to call them back.’ (The gender of the lawyer isn’t important)

If you wish to refer to abstract person in an impersonal way, you can use "it". This is nothing new. Sometimes, this is the first choice, for example: "In the dim light, I noticed the figure of a man moving in the distance. It was walking slowly." In your example, "it" seems to me like a natural choice because the "child" is an abstraction, not an individual. It would be insulting, however, to refer to a specific child as "it". For example, you should never say to the parents, or anyone else "Billy misbehaved, so I told it to write 'I will be good' a hundred times on the blackboard."

It seems to me that the phrase "a child" cues the pronoun "it." "A child" is understood to stand for the whole category, but grammatically it's singular. You could say "to allow a child to do everything that it wants to" or "to allow children to do everything that they want to." But the use of "they" as a singular has been legitimate for centuries, and it has become more popular as the best gender-neutral pronoun. Referring to "a child," the problem is choosing between "it," "he," and "they." "It" is unpleasant because "it" is used with nonliving objects. A couple of centuries ago people may have really understood "he" to include both genders, but now it is considered disrespectful to females. And the singular "they" still "sounds wrong" to some.

Yes. This is a somewhat controversial and ever changing topic. You will find examples of the use of "it" to refer to a child when the gender is unknown. Especially in older examples. There is an argument that "they" should be used to refer to an unknown subject. I believe"they" is more common in modern English, and indeed this is what we now usually teach. Partly because if you saw some parents with a baby (and you didn't know the gender) and you said "It is cute." they might be a little offended and say "She" or "He". Partly because this fits with what Examiners expect. Using "it" in this way isn't wrong... and to be honest there is no Grammar Police Authority that will check your grammar... but I advise you to use "they" in type of situation. Just make sure to remember verb agreement. Also, of course there is also a growing trend for people to not identify their gender and to use "they" for singular subjects who are known. So you will sometimes hear"They are called Chris." instead of "He/she is called Chris."