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Hi i believe i have a considerable grasp of the english language and its use. My pet hate is the incorrect use of spoken english.

I believe the word "nothing" is often used in sentences when in fact the correct word to use is in fact "anything".

Here are some examples or incorrect useage of the word Nothing:

"Yesterday we did not learn NOTHING in class.": i believe the word nothing should be replaced by the word anything

"You did not do nothing all day": i believe the word nothing should be replaced by the word anything

"You have not understood nothing that i have said": i believe the word nothing should be replaced by the word anything

"I did not do nothing wrong": i believe the word nothing should be replaced by the word anything

Can you please advise if my belief in the correct use of the word anything as in above examples is correct and what is the actual grammatical rule that needs to be applied to determine when one uses the word nothing v's anything in a sentence.
Comments  
The words nothing, nobody, nowhere are already negative.
Consequently, you don't use them with "not".

Examples:

I do not know anybody in my apartment building. = I know nobody in my apartment building.

I don't know anything about it. = I know nothing about it.

You are correct, then.

Yes. English never uses double negatives to negate. However, Spanish (and some other languages) do use double negatives.

"No esta nadie." Literally: "there isn't nobody." But it MEANS "There isn't anyone."
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In Portuguese is the same:

"Não há ninguém" = There isn't nobody.
My pet "hate" is people who do not capitalize the word "I" repeatedly in a sentence.