Whether ‘he’ or ‘him’, the message is the same. It just depends upon whose translation one is using. I’ve seen it quoted both ways.
It isn’t the same. Leave out the relative clause “who is without sin” and just keep the main clause.
The choice is between “Let he throw the first stone” or “Let him throw the first stone.” “Let him” is obviously the grammatically correct choice. That is because him is the direct object of let.
Let “he” isn’t English. Doesn’t anyone know basic grammar? Nominative and objective cases?