What is the ruling on touching one's female cousin—who is a stranger (non-mahram)—out of fear for her while she is riding a motorcycle, and is expiation (kaffarah) obligatory for that?

What you did is a reprehensible act, a great door to temptation, and the temptation of women is immense. That is why the Sharia warned against it and commanded lowering one's gaze from looking at them, in affirmation of the Almighty's saying: "Tell the believing men to lower their gaze and guard their private parts. That is purer for them. Indeed, Allah is Acquainted with what they do." If this is merely about looking, then what about physical contact? It is authentically reported in Al-Bukhari and Muslim, on the authority of Usama ibn Zayd (may Allah be pleased with both of them), that the Prophet (peace and blessings be upon him) said: "I have not left behind me any temptation more harmful to men than women." And in Sahih Muslim, on the authority of Abu Sa'id Al-Khudri (may Allah be pleased with him), the Prophet (peace and blessings be upon him) said: "Indeed, the world is sweet and green, and indeed Allah is going to make you successors in it, and He will see how you act. So, fear the world, and fear women; for indeed, the first temptation of the Children of Israel was in women." And also in Al-Bukhari and Muslim - and the wording is that of Muslim - on the authority of Abu Hurairah (may Allah be pleased with him), the Prophet (peace and blessings be upon him) said: "The son of Adam has been allotted his share of adultery, which he will inevitably commit. The adultery of the eyes is looking, the adultery of the ears is listening, the adultery of the tongue is speaking, the adultery of the hand is seizing, the adultery of the feet is walking, and the heart desires and wishes. And the private parts confirm that or deny it." What is obligatory upon you is sincere repentance for what you have done, and the conditions of repentance are explained in Fatwa No. 46149.

Our advice to you is diligence and striving, and to follow the path of righteousness and steadfastness in obeying the Lord of the Worlds, and to take all measures that help you in that, such as sincerity and devotion, accompanying the righteous, and attending circles of knowledge.