Badi'ah
Yes. Badi'ah is formed from the classical Arabic adjective بديعة and is linguistically authentic with roots in Arabic lexicon.
No. The proper name Badi'ah does not appear as a proper noun in the Quran, though the root b-d-ʿ appears in classical Arabic usage.
It carries a literary, classical tone and is often chosen for its elegant meaning and Arabic linguistic purity.
Badi'ah is rare. It is known among scholars and those favoring classical Arabic names but is uncommon in general population registers.
Related forms include the masculine or neutral root-based usages like Badi' (بديع) for males or the adjectival uses; however, Badi'ah is specifically feminine.