Is it permissible to gift the book "Ar-Raheeq Al-Makhtum" (The Sealed Nectar), which contains Arabic Quranic verses and their English translation, to an elderly Christian woman, and if not, what is an alternative book in English about the biography of the Prophet (peace be upon him)?

If it is possible to guide the elderly Christian woman to an electronic copy of the Book, that would be preferable. If that is not possible, and it is highly likely that she will not disrespect the Book, then there is no harm in giving her a paper copy in English that contains Quranic verses, prioritizing the benefit of her دعوتها (invitation/call to faith) and her faith. Some scholars permit giving books of knowledge to a disbeliever, and the correct view is that it is permissible absolutely, as long as it is not a Mushaf (a complete copy of the Quran) or something equivalent to it.