Is the punishment mentioned in the noble Hadith, "Any woman who asks her husband for divorce without a sound reason, the scent of Paradise will be forbidden to her," incurred by merely the woman's request for divorce, or by the actual occurrence of divorce after the wife's request?

The warning of Paradise's fragrance being forbidden to a woman who seeks divorce without a valid reason applies to the mere request, not to the actual occurrence of divorce. What mitigates the severity of the punishment is the qualifier "without a valid reason." So, if a woman dislikes her husband's character, or his physical appearance, or if he is deficient in religious practice, or if he withholds her right to financial support, or if he does not give her her right to conjugal relations, then she has the right to seek khul' (divorce initiated by the wife). The warning is sufficient to make this a major sin.