When we use a name with its article then it is definite, I cannot make the following sentence more general. I’m trying refer to all children in the world. And why do we use the pronoun es in such sentences? Is my second sample also correct?
Die Frau in meinem Büro kann es nicht ertragen Kinder.
Die Frau in meinem Büro kann Kinder nicht ertragen.
And in the following example why is -es droped, shouldn’t it be: meines Kind because Kind is neuter?
Ich kann mein Kind nicht mehr ertragen.
The rule is rather simple: To make definite references, use the definite article. For indefinite references, use the indefinite article in the singular or no article in plural.
Let’s strip your examples of everything unnecessary:
Sie erträgt Kinder nicht.
This one has no article, so it generally means she cannot stand children.
Sie erträgt die Kinder nicht.
She only cannot stand these (previously established or clear by context) children.
Moving to singular:
Sie erträgt ein Kind nicht.
Simply Kind would be wrong, and the sentence altogether is weird, because why would you only not stand one child?
Sie erträgt das Kind nicht.
She cannot stand the child (clear by context or previously established).
Of course, further context can be given with adjectives if desired:
Sie erträgt schreiende Kinder nicht.
Sie erträgt ein schreiendes Kind nicht.Sie erträgt die schreienden Kinder nicht.
Sie erträgt das schreiende Kind nicht.
(Here the indefinite singular does not sound as bad as above.)
The first sentence doesn't make sense at all. One might understand it as
Hey kids, the woman at my office can't stand it.
But in German, there'd be a comma missing before "Kinder"...
The second sentence is fully correct. It means:
The woman at my office can not stand children.
The "es" in "kann es ertragen" is only required if there's no closer description of the required accusative object. For example:
Ich kann das Wetter nicht ertragen. (No es here, as das Wetter is the accusative object)
as opposed to.
Wie findest Du das Wetter? Ich kann es nicht ertragen. (es stands for das Wetter here and is the accusative object)
The last sentence is really a matter of understanding the accusative case:
Nominativ: mein Kind
Genitiv: meines Kindes
Dativ: meinem Kind
Akkusativ: mein Kind
Ich kann [wen oder was, Akkusativ] nicht ertragen.
Ich kann mein Kind nicht ertragen.
Your examples 2 and 3 are correct, 1 is wrong. A legal example similar to example 1 would be:
Die Frau kann es nicht ertragen, Kinder um sich zu haben.
Meines would only be valid in genitive singular, like:
Der Mantel meines Kindes ist verschwunden.
