I and others have stated above and elsewhere that the common usage in English is for the abstract negative to be plural. So "We don't have children" should be accepted, as that's how you would say it in English, unless you had a specific child with you, who had since run off, but then you would use "the child", not "a child". We want to get the English right as well as the Russian.
English has two different sentences "We don't have a child" and "We don't have children". So does Russian. The second is more common in English, and I believe in Russian too. The first, less common, sentence is given here, the corresponding English sentence should be used in the translation.
Masculine nouns ending in a consonant add -а in the genitive case. The "о" in "ребёнок" is what's called a "fleeting vowel" - when words end in an awkward consonant cluster it's broken up by adding -о- or -е- in the middle, so the nominative form is "ребёнок" instead of "ребёнк". In cases other than nominative the fleeting vowel is dropped out so "ребёнок" changes to "ребёнка".
"The child" refers to a specific child. If you're saying that you're childless, then you're not talking about a specific child, so it would be wrong to use "the". I guess you could say that if you meant that a particular child was not with you at the moment, but I'm not sure if the Russian can be interpreted in that way.
Even though Russian has a singular noun here, it is not correct English grammar to say "We do not have a child." Shouldn't real sentences and real used be more rellevant than the gramatically most acurate? After all, the main point is to learn languages, not to make some pointless translation that is confusing and useless.
This was discussed above. It's actually perfect English grammar to say "We don't have a child". But it's a lot more common to use the plural "We don't have children". And it's the same in Russian. So it would actually be misleading to translate this sentence as "we don't have children" - both because it would be confusing to beginners to translate "ребёнка" as plural "children", and because it would give a false impression about the Russian sentence.