Basīma
Yes. Basīma derives from the Arabic root b-s-m (ب س م) and is the feminine active participle meaning 'smiling'.
Yes. Basīma is used by Urdu-speaking Muslim families and retains the same meaning in Urdu.
No. Basīma is a valid Arabic word and personal name but it is not a Quranic proper noun; quranic_reference is empty.
Basīma is attested as a given name in Arab and South Asian onomastic records but it is relatively uncommon; it is not widely associated with a single famous historical figure.
In Arabic script it is written باسمة. In Urdu script it is written باسمہ or باسمة depending on regional orthography.