I, too, was a child, and so I have my own counter-anecdotes. Most taunts from kids don't have wider cultural implications. Sexist insults are an exception--as are racist insults, you certainly wouldn't be so dense as to claim that a kid calling another black kid a nigger derives its power from 4th-grade-power-dynamics would you?
That depends on the context. Believe it or not, plenty of kids of all colors have no idea what nigger means or why it is offensive. A white kid calling a black kid a nigger can be racism, or it can be that the kid has no idea what it means and also calls white kids and asian kids niggers too. That's really the very crux of the issue here, sexism is a motivation, it doesn't exist purely because the word sandwich is used. Just as the word nigger does not automatically mean racism is involved, racism is a motivation that could cause someone to use the word, but that doesn't mean you can extrapolate backwards and say there is always racism behind the word.
Your defense of your point is bordering on the absurd. Yes, its technically true that these things may not indicate racism/sexism is the motivation behind their usage, but the vast vast majority of their usages are derived from racist/sexist thoughts from the wider culture or in the individual themselves. This is even more true in the case of the word nigger. To offer your point as if it has any usefulness in practice is ludicrous.
