Eimaan
Yes. Eimaan is derived from the Arabic noun إيمان (īmān) meaning faith; the spelling 'Eimaan' reflects Urdu/Persian-influenced transliteration.
Yes. The root and the noun إيمان appear multiple times in the Qur'an; a clear instance of the concept occurs in Surah Al-Baqarah 2:285 (contextual forms of belief).
Yes. The orthographic form 'Eimaan' appears in Urdu transliteration and in modern South Asian Muslim naming practices and literature as a feminine virtue-name.
The underlying noun īmān is gender-neutral as a concept; as a given name 'Iman' or 'Imaan' can be used for men in some cultures, but 'Eimaan' recommended here is presented as a feminine form.
Pronounce it as ee-MAHN (IPA: /iːˈmaːn/), with stress on the second syllable and a long initial vowel.