I figured it was just because she was a natural redhead. Although as Reverend says, it's probably exaggerated for aesthetic effect, reminiscent of how the New Batman Adventures Catwoman had pure white skin when she was in costume (though not in her civilian look, a difference that was never explained).
It's in the Arrowverse. We know Bruce Wayne exists on Earth-1 because Oliver name-dropped him, and it's been all but explicitly stated that Batman exists on Earth-38 (Superman's vigilante friend with lots of gadgets, lots of baggage, and something pointy sticking up from the sides of his head).
Besides, it's not a "synopsis," it's just a brief, initial press announcement, way too cursory to treat as definitive. We can't make firm conclusions about any aspect the show based on a measly couple of sentences. Obviously we'll find out for sure one way or the other once we actually see the show; until then, the sensible thing is to keep an open mind.
Well, Batgirl was always more directly linked to Batman, so maybe Huntress felt that Batman had the right to decide who got to wear the mantle. Kate Kane may have been inspired by Batman to become a vigilante in a similar way to Barbara Gordon, but she chose to pursue it more independently. I gather that Batman did disapprove of her using the imagery at first, but she just refused to let him dictate to her what she could do, so he had to grudgingly accept it.
