any has several meanings. Two of them are:
Do you have any sugar? - any means an indefinite quantity
You can choose any pizza - any means "it doesn't matter which".
Looking at your first example:
I like dogs. - We are using the plural of dogs to mean "dogs in general"
I like all dogs - By adding all we really mean "all dogs". It's emphatic.
I don't like dogs - Back to "dogs in general"
I don't like any dogs - any means "it doesn't matter which"- It's emphatic.
You can emphasise both sentences further by adding "at all", though the second sentence, with double emphasis, is uncommon.
I don't like dogs at all
I don't like any dogs at all
With the second example, children, we are talking about an indefinite quantity. This meaning does not give any emphasis, so there is no difference between these two sentences:
Do they have children?
Do they have any children?
The reply could be one of the following: again, "any" does not have an emphatic effect, so there is no difference between the two negatives.
Yes, I have children
Yes, I have x children
No, I don't have children
No, I don't have any children
You can make the negative sentences emphatic by adding "at all" as we did with the previous example:
No, I don't have any children at all.