Ibtihaj
Yes. Ibtihaj is Arabic in origin, derived from the verbal-noun pattern of the root ب-ه-ج and means joy or delight.
No. The noun ابتهاج appears in Arabic vocabulary, but the specific personal name Ibtihaj is not cited as a proper name in the Quran.
Yes. A well-known contemporary bearer is Ibtihaj Muhammad, an American Olympic fencer, which demonstrates modern attestation of the name.
It is relatively rare compared with many traditional Muslim female names; it is used in Arabic-speaking communities and among Muslims in Bosnia and elsewhere.
Common informal shortenings include 'Ibtih' or 'Hajah' in colloquial use, but many families prefer the full form because of its positive meaning.