Chand Bibi
Yes. Chand Bibi is historically attested as the name/title of a late 16th-century Deccan regent and leader; the compound combines regional Persian/Urdu elements rather than being an invention.
No. 'Chand' is the Persian/Indo-Aryan word for 'moon' used in Urdu/Hindi, and 'Bibi' is a Persian/Urdu honorific; the name itself is not from the Quran and has no Quranic ayah as its origin.
The compound form 'Chand Bibi' is primarily South Asian (Deccan/Bengal/Mughalate contexts) and is rare outside regions influenced by Persianate cultural practices.
Yes; it is culturally rooted and religiously neutral (not taken from scripture). Families who appreciate historical or poetic names often choose it.
Related moon-themed feminine names in South Asia include Chandni and Chanda; 'Bibi' is an honorific that can combine with other personal names.