The full name of Rachel Skarsten, who plays Alice, is Rachel Alice Marie Skarsten. Marie is also Beth Kane's middle name in the show.
Batwoman clearly has black paint around her eyes whenever she wears the cowl. But whenever she takes the cowl off, there's never any paint around her eyes. This happens in several episodes and is common amongst many superhero/comic book shows with masked characters.
The Warner Bros and DC Comics logos are colored red and black (Batwoman's colours), are set in Gotham City and have bats flying around.