The sentence may be loosely translated as "They have xxx children".
It has nothing to do with numerals, it is so known "Безличное предложение" (impersonal sentence). There is no explicit subject in the sentence. Such sentences imply that what happened, happened without consent of the objects mentioned in the sentence. For example "Я не спал" implies "I didn't sleep because I choose so" while "Мне не спалось" means "I didn't sleep because I could not.".
Impersonal sentences must not be confused with 'truncated' sentences, when either the subject or the verb is omitted because obviously derivable in context.
Impersonal sentences may be very loosely paralleled with "there is/are" construct: where is that there? Same here, there is no "who" or "what" that "родилось" in this sentence.
Yes, it is a case when difference in one symbol radically changes the way the sentence must be parsed (though, luckily, in this case the meaning does not differ much).
As for the second part
What about the second one...would it be wrong to write "У них родилось два
ребёнка?" insted of "двое"? And why is детей in genititve plural and not in
There is some freedom here.
The proper plural form of "ребенок" is "дети" (very much like forms of "to be" in English),so it is possible for uniformity use "двое детей" instead, yet it is possible to use "ребенка" for numerals below five . A bit more consistent example would be "гусь". One can say "Один гусь, два гуся, три гуся, четыре гуся, пять гусей, шесть гусей, .... двадцать гусей", but also "Один гусь, двое гусей, трое гусей, четверо гусей, пять гусей, шесть гусей, .... двадцать гусей" (for non-round numbers above 20 the inflection is selected on basis of the last word in the combined numberal)