Does sincere repentance avert punishment from Allah, and does her husband have rights he should claim from her on the Day of Judgment, and what should she do to please Allah Almighty?

Your friend must conceal herself and not tell anyone about the fornication she committed. If her repentance is sincere, Allah will forgive her and transform her evil deeds into good ones, as stated in Surat Al-Furqan. You must be a support for her in maintaining righteousness. As for her husband having committed fornication, this does not concern her, and she is not permitted to utter it lest she fall into the sin of slander.