So even though this is set in a world with people who have super abilities magical abilities, there is Amazons, Aliens, an entire society that lives underwater, and talking gorillas and it is not that all strange for any of these people to dress up in costume and be heroes and villains and yet the military still cares if your homosexual.
Like Reality Unless Noted. Besides, half a century ago people would wonder about our own world, where we've been to the moon, have cloned animals, split the atom (Do people still say "split the atom" when referring to nuclear technology?) and created the technology for everybody on the planet to be able to contact everybody else at will, and we still care if homosexuals are in the military (Hopefully not for much longer). People have shown a remarkable ability to hold onto their prejudices no matter what is happening in the rest of the world.
And if you're implying that the presence of superhumans would make gays seem normal by comparison...well, the superhumans aren't openly serving in the military, either.
The military is trying it's hardest to do so, tough.
After seeing the preview page from her ongoing series, something bothered me: she's a little too pale. Pale is fine, but then again, there's ginger, and then there's Ronald McDonald. She just looks a little Uncanny Valley-ish standing next to others. Is there an in-story reason for this? Skin bleached by weird chemicals? The Joker's sister?
Might be Reality Is Unrealistic, since real redheads can actually be that pale (in fact, the person writing this is such an example), since they're physically incapable of producing the enzyme that causes people to tan. That aside I'd guess it's an artistic choice, since the model sheet of her was described as having "vampire porcelain white" skin. And check out this photo◊ of Nicola Roberts of Girls Aloud if you think Kate's unrealistic.
Sheamus of the WWE is another individual in real life who is both a redhead and so pale that some fans new to the WWE product will and actually have done a doubletake the first time they see him. This troper included, despite it being a case where he was new to the product instead of me being new to it, and I still did a doubletake the first time I saw him. He seems so pale that it borders on the idea that many might think that it almost has to be special effects makeup, but no, he really is that pale. So Kate's paleness in fiction isn't too surprising, as both the above example and this one should hopefully prove.
One of the reasons I believe Rucka himself suggested during Batwoman's appearances in Detective Comics was the psychological trauma of her experience at the hands of the Religion of Crime. That said, that doesn't quite mesh with the fact that her new appearance is used in the 'Go' origin story. It could also be the fact that she's largely running around at night and - as the joke often goes with Batman - sleeps during the day, meaning she doesn't get much sun. It's also likely a stylistic choice, matching her rockabilly/goth garb.
When she wanted Bette to stop being a superhero due to the risks involved, Kate told Bte that she just didn't have what it takes. Why not be honest with her and tell her that it's too dangerous? Is there any reason for Kate not to be honest that I missed?
Because Kate was a soldier and thinks like a soldier. she probably belives a blunt "you are not made for this" is more effective to make her stop than "it is too dangeorus for you to do this". she is basically meaning the same thing, that bette could get killed, but the way she expressed it is because of the way Kate thinks, like a soldier with a cadet.
Are the Kanes and Waynes related?
Yes, through marriage. Martha Wayne, Bruce's mother, was originally Martha Kane; her maiden name had been established decades ago. Kate and Bruce specifically being cousins was established during the current series, whereas previously it had just been nebulous and could theoretically have been a naming coincidence.