"I have six children."
I believe 子供たちが六人です。 would refer to six groups of children. たち is a plural particle that makes groups if I remember correctly.
It isn't necessary to pluralize usually. In this case the plural can be inferred from the fact that there are six of them.
Does the audio sound more like に than にん here? I'm hearing 六にいます but could just be my beginner's ear.
です is copula, to be.
います is the closest to "there is" for living things.
あります also means "there is" but for things considered not living.
Iru and Imasu are the casual and polite way of saying the same verb "to be" (for living things).
Is adding 私は (or わたしは) to the beginning of the sentence correct here? It's been my understanding that 私 is the implied subject here, but it can be added explicitly if desired. I tried adding it in the website interface but I get a mistake. Wanted to flag "my answer should have been accepted" but I don't really know if I'm right.
子供は六人います。 There are 6 kids.
子供が六人います。 I have 6 kids. "I" is implied as the topic. Sooo, 私は子供が六人います。
based on my understanding (I'm also a learner so take it with a grain of salt) this wouldn't be correct because 子供 is not the main subject of the sentence but the implied subject 私 (わたし). は is usually used for the primary subject of the sentence. In your sentence the children become the main subject so the sentence is more accurately translated "there are six children" (that's also how google translates it, although it also translates the original sentence from the exercise the same)
私 is background information that would be indicated by the topic particle. I'm sure that's what you meant, but unless there's a conjunction, there wouldn't be two "subjects" in a sentence.
You can think of it as "As for (x)" when translating a phrase "x は." "As for the children, there are six of them" vs. "As for me, there are (I have) six children."
Short answer: Because the words are completely disarranged.
Long answer: Maybe you would translate it like "My people six children are." - makes no real sense. In Japanese "(living being[s]) が (number) います" means (besides other meanings) "there are (number) (living being[s])". Number is optional. So "子どもがいます" could mean "I have a child" / "I have children" or just "There is a child" / "There are children". Here a more clear translation would be "私は子どもが六人います" ("I have six children"). But it depends on situation. If someone asks you about your children, it would be clear that you talk about them, so "子どもが六人います" would be a clear answer to that question. Hope I cleared things up somehow :-)
"There is/are" and "I have" are kind of same word in Japanese - います (animate objects) / あります (inanimate objects). That's not 100% exactly the whole story, but in many situations where you'd use "I have" in English, you'd use います / あります in Japanese, like in this example.
I believe that you can move around the terms fairly freely as long as you move the associated particle with it. That said, I believe verbs stay at the end, and the topic that is indicated by the topic marker stays at the beginning.
because you are not speaking about your children but about how many children you have. You can however say "私は子どもが六人いる" but its a little redundant
try 六人の子どもがいます, seems like duolingo doesn't accept this structure without the particle.