Anbu black ops are a special unit of shinobi. While Anbu is not technically one of the primary ninja ranks, most Anbu ninja were first jonin before they were recruited into the Anbu black ops. So in a sense, yes, they are generally more powerful than a regular jonin.
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The ANBU you are talking about is Yugao Uzuki, the lover of Hayate Gekko, a leaf Jonin.
no
As of the start of Naruto Shippuden, Neji is a Jounin. In the first series though he was a genin like all the others minus Shikamaru.
Jonin is the rank under Kage/Village head. Before jonin comes chunin.
the only female ANBU Black ops member in the anime and manga was Hayate Gekko's lover, Yugao Uzuki. In the game Naruto: Clash of Ninja 2, there are two other female ANBU members named Kagura and Komachi.