>   1. こども **が** ねて いますから、しずかに して ください。
>
>   2. こども **は** ねて いますから、しずかに して ください。
>
>

I know that the 2nd one is incorrect. But why? To my understanding, が is used
to emphasize the subject we can hear or see, while は is rather for emphasizing
the subject with description or judgement. So it seems that both are correct
to me.
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particle-は particle-が は-and-が
こども "が" ねて いますから
As far as I know, it is usually never technically incorrect to replace one for
another, but the emphasis will be different and so will the meaning.

Using は in your example, would give me this impression.

> Maybe YOU are not sleeping, but THE KIDS are, so calm down!

With the が the emphasis on comparing the sleeping people and the non sleeping
people is gone, and it becomes much more like a factual statement, which is
usually what you wish to say.

> [just so you know] the kids are sleeping so please don't make too much sound
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A useful rule to follow is this:

は indicates that what precedes it is the topic of the whole of the rest of the
sentence up to the next full stop

が marks the subject of the next verb that is in a sentence-ending form (and of
any intervening verbs in non-sentence-ending forms) but not of any subsequent
verbs in the same sentence.

In other words, は lasts for the whole sentence, but が is 'used up' by the next
sentence-ending form, so any subsequent verb in the same sentence must have a
different subject.

Try it out on these two:

こどもがねていますから "The child is asleep, so . . ." (こども is now used up, so the next
verb will have a different subject)

しずかにしてください ". . . please be quiet" (subject is obviously the person addressed)

こどもはねていますから "The child is asleep, so . . ." (こども will be the topic/subject of
the next verb so that verb can't be a request such as しずかにしてください: sentence 2
is therefore invalid. The sentence could perhaps conclude with something like
ないていません, giving "The child is asleep, so it isn't crying")

The sentences traditionally used in Japanese school textbooks to illustrate
this point are:

とりはなくときくちをあけます "When birds sing they open their mouths" (とり is the subject of
both verbs: "Birds [topic] - when they sing they open their mouths"

とりがなくときくちをあけます "When birds sing I open my mouth" (Subject とり marked with が is
used up by verb なく, so あけます must have a different subject, and in the absence
of a new subject being provided we fall back on default "I".)

Consider also these examples:

ちちはびょうきになったから、いそいでかえった "My father, because he fell ill, hurried home"

ちちがびょうきになったから、いそいでかえった "Because my father fell ill, I hurried home"

おとうとはらくだいしたからだいがくをやめた "My younger brother failed his exams so he dropped out
of university"

おとうとがらくだいしたからだいがくをやめた "My younger brother failed his exams, so I dropped out
of university "
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