How will you learn the difference between "enfant" and "gamin(e)" in French if we accept both "child" and "kid" in English?

1) Yes, there is. "lady" is much better translated as "dame".
2) A baby goat is "un chevreau" (note that even though the grown goat "la chèvre" is feminine, the baby is masculine). "kid" in reference to humans is "gamin(e)". So no, they're completely unrelated in French.

No, "kids" is "gamin(e)s".

"les hommes" = "the men"
"ont" = 3rd person plural form of "avoir" (to have)
"des enfants" = "(some) children"

So, "Les hommes ont des enfants" means "The men have children". "have got" is an informal way to say "have" in British English.

"ont" is the third person plural (they/ils/elles) form of "avoir" (to have).

No, for "the people" we would use "les gens" in this context.

Also, nothing in this sentence says they're a gay couple. It could be talking about the men at a father-son group, for example.

These are different conjugations of the verb “avoir” (to have).

Use the 1st person plural to talk about a group, which you are part of: we have – nous avons.

Use the 2nd person plural to talk about a group, which you are also talking to in this moment: you (guys) have – vous avez.

Use the 3rd person plural to talk about a group, which neither you are part of, nor is the addressee of your speech: they have – elles/ils ont.

The formal addressing of people is to call them like if he/she was a group: you (sir/madam) have – vous avez.

Find an overview of this topic in the tips and notes of the skill “Basics 3” (click the light bulb), only available in a web browser, not in the Duolingo app: https://www.duolingo.com/skill/fr/Basics-3/tips-and-notes

When followed by a vowel, it's called liaison

If you want to point out the “some”, then you can say so in English. But the French “des” doesn't point out any concept. “des” is the usual indefinite plural article and it has to be there. Whereas in English there is no indefinite plural article, instead you place a so-called null article, which in fact is just nothing.

To sum it up, you have to put “des” and you should omit “some” in most cases. The given translation in Duolingo is fine.

Also read the tips and notes of the skill “Basics 2”: https://www.duolingo.com/skill/fr/Basics-2/tips-and-notes

“These” / “Ces” is a demonstrative article and is used to point at certain men, who are not known from context.

“Les” / “The” is a definite article and denotes the certain men are known from context, but it doesn’t point at them.

For reference, please see the tips and notes of the skill “Demonstratives 1”: https://www.duolingo.com/skill/fr/Demonstratives-1/tips-and-notes

Ils ont = They have Ils sont = They are

See the full conjugation table: https://www.duolingo.com/skill/fr/Basics-3/tips-and-notes

If you read IPA, "ont" is /ɔ̃/ and "un" is /œ̃/. The main way to tell them apart is practice - I recommend listening to them side by side on Forvo.com or even Google Translate.

Context is also useful, there aren't that many situations where either would really fit in the sentence.

"infant" is "bébé" in French. All "bébés" are "enfants" but not all "enfants" are "bébés".

I'm afraid we can't see what you wrote or the correction you were given, so we aren't able to help unless you give us your exact sentence in full and the exact correction.

The underlined word is often not where the problem actually is, so the error may well be elsewhere in the sentence.

Because LES is an indefinite article and THE is a definitie article, they are not equivalent.

The men have the children = Les hommes ont les enfants

The men have (some) children = Les hommes ont des enfants