Laugh can be used as a noun, but in that form, it's usage needs to be defined by an applicable form of grammatical structure such as use of an article, like you used in your preface (a laugh).
Laugh can be used as a singular or plural with the right construction, but your sentence examples don't achieve internal agreement, especially your original example that is now example #2.
Your new example #1, I like the sound of this laugh of the children, is very awkward, but it could be interpreted in a way that is technically correct, although the meaning would be different from what I think you intend.
For example, if you had a group of children who all had a similar distinctive laugh and you liked the sound of it, that sentence might be appropriate; "this laugh" would refer to a singular type of laugh shared by all of those children. But that's the only way that sentence would be correct.
There is no way that example #2, I like the sound of children's laugh, can be correct. In that construction, laugh has no meaning or usage that is consistent with plural possessive "children's" and the absence of a defining element like an article.
If you were to add "the" — I like the sound of the children's laugh — you could interpret it in the same odd way as the previous hypothetical example (and that would be a less awkward way to do it). But that's not the meaning you're trying to convey.
So how can you integrate laugh and children into a sentence with the kind of meaning you describe in the question?
You could say, I like the sound of a child's laugh (note the article and singular use). That's internally consistent and conveys the concept.
If you want to use the plural children, you can't use laugh, at least not in a sentence construction similar to your suggestion that I can think of, because in that context, laugh would be singular. Children laugh, but that's the verb form.
In your sentence, you could use, I like the sound of children's laughter or you could also say I like the sound of children laughing (verb in this case).