Some sentences
1. Rules are definitely important for not only children, but also (for) the whole society.
Q1: is 'for' necessary in this sentence, why?

2. The main function of rules is for children to understand what they should and should not do, to tell them right from wrong.
Q2: is 'tell them right from wrong' a right expression?

3. They depend on others to tell them what to do all the time so that they can’t live on their own after they grow up.
Q3: is 'so that' properly used here?

4. I agree that strict rules may have adverse impacts on children; nevertheless, we can’t throw away the apple because of the core.
Q4: why there is a semicolon before nevertheless?

Question

Some sentences
1. Rules are definitely important for not only children, but also (for) the whole society.
Q1: is 'for' necessary in this sentence, why?

2. The main function of rules is for children to understand what they should and should not do, to tell them right from wrong. 
Q2: is 'tell them right from wrong' a right expression?

3. They depend on others to tell them what to do all the time so that they can’t live on their own after they grow up.
Q3: is 'so that' properly used here?

4. I agree that strict rules may have adverse impacts on children; nevertheless, we can’t throw away the apple because of the core.
Q4: why there is a semicolon before nevertheless?

Answer

To translate your “apple because of the core” idiom, I’d use the English idiom “(don’t / we shouldn’t) throw out the baby with the bath water”:

https://idioms.thefreedictionary.com/throw+out+the+baby+with+the+bath+water

https://en.wikipedia.org/wiki/Don't_throw_the_baby_out_with_the_bathwater

Answer

1. The for is uneccesarry in the first sentence. That's all I have to say for the moment.
Yaminou AKODA

Answer

These are all poor sentences.
1. Rules are important not only for children but for society as a whole.
2. You “teach” children right from wrong, not “tell” them
3. “So that” is incorrect. Correct would be “and as a result (without the second “they”)” or “and consequently”. “So that” means you that you do something to intentionally to produce a good result. “He studied hard so that he could pass his exam”
4. “We can’t throw away the apple because of the core” isn’t good. It’s not an expression but if you wanted to express this thought using this idea (which I wouldn’t), you might say “you don’t throw away an apple because it has a core” or “you don’t throw away the whole apple because it has a core”

Answer

1) yes. Why ? - because it is.
2) yes
3) yes , correct. Also, 'can't' is the correct word if you mean someone is so dependant on others that they don't know what to do unless someone tells them.
4) the use of semicolons is controversial ; in this sentence it is correct, but not the only correct way of writing the sentence.

Answer

Q1. 'For' is grammatically necessary in the sentence because it is serving the purpose of linking the adjectival phrase 'not only for children' with the noun phrase 'rules are very important'.

Q2. 'Tell them right from wrong' is a correct expression.

Q3. 'So that' is properly used in the sentence. Be careful with contractions. I'm fairly certain you meant to write 'can' instead of 'can't'

Q4 The purpose of semicolons is to link to major sentence elements, provided there is no coordinating conjunction in the sentence. So in this sentence, the use of the semicolon is correct.
And, but, for, nor, or, so, and yet—these are the seven coordinating conjunctions.