Does informing someone of sad news—thereby causing them psychological harm—constitute an injustice requiring the forgiveness of the wronged person, or is it merely misbehavior that does not amount to injustice? And what if this behavior is repeated after advice has been given?

It appears that your cousin's action was not intentional harm, and informing someone of sad news in that manner is generally permissible and usually does not cause harm. Therefore, it cannot be said that merely informing someone of sad news is forbidden. Consequently, she is not required to seek your forgiveness. However, it becomes forbidden if it is known that the information will be harmful to the recipient and the intention was to cause harm, because intending to cause harm without right is a forbidden injustice, as the Prophet (peace be upon him) said: "There is no harm nor reciprocal harm." Thus, the prohibition hinges on knowing the harm and intending to cause it. We advise you to forgive, pardon, and overlook what your cousin did.