What should one do who opened his wife's will after she asked him by Allah not to open it until after her death? How can he atone for that, and should he inform her or not, and how grave is his error?

You must repent to Allah Almighty for revealing what your wife asked you to keep secret until after her death, as this is a form of betrayal of trust, and also for not answering her when she asked by Allah, which is a detestable or forbidden act. The conditions for repentance are well-known, and the original principle is to seek your wife's forgiveness. However, if a greater harm is feared from doing so, then abundant supplication for her and seeking forgiveness is sufficient. This suffices for repentance from such moral rights, and there is no specific expiation for this.