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 Bette 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 believes a blunt "you are not made for this" is more effective to make her stop than "it is too dangerous 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.
So you have a gothic-looking woman with milky white skin and red hair and a dark costume vigilante with white skin and red hair and no one makes the connection? How many white skin gingers are in her city?!!
The hair Kate has while she's Batwoman is a wig attached to her mask, which would get noticed and assumed to not be Batwoman's real hair color. Same for the pale skin; as far as the public is concerned, who knows what all these costumed freaks put on at night? Pale makeup wouldn't be too outlandish. Also, Kate's public persona is the Rich Idiot with No Day Job and she's relatively well-known in Gotham, so she's safe in that regard, too.
How exactly does Batwoman (and Batgirl for that matter) manage to fight multiple hordes of enemies without a single one grabbing her long flowing hair, slamming her face into the concrete, and beating her to death? One of the most basic things you learn in training is to take any handholds on your enemy that you can, and hair is one of the best handholds in existence. So why does Batwoman not only allow her hair to flow as long as it does, but also get away with it in every single fight? Unless she has the logic-defying powers of Squirrel Girl, she should have died during her first fight. And even if it really is just a wig attached to her mask, either one pull and her secret identity is exposed, which is arguably just as bad, or we have the same handhold problem we just mentioned. In either scenario, Batwoman loses.
Batwoman's mask coming off does become a plot point in one of the later arcs in her ongoing series and ends up being a serious problem for her, and the issue is brought up by Batman during Batwoman's appearance in Detective Comics (which is where it's revealed the long hair is a wig). One could argue that Kate (and Babs for that matter) are simply good enough fighters that any disadvantages caused by the hair are made irrelevant by their skill. Kate in particular is on the level of a Green Beret or Navy SEAL at the very least. Is it the wisest decision, letting their hair hang loose? No. But it's not like grabbing someone's hair in a fight automatically means they lose, either. You could just as easily make an argument that none of the Bat-Family should wear capes for precisely the same reason.
That makes no sense even for a comic book. Aside from the fact that Batman's cape has a tear-away feature for just that purpose, Batwoman and Batgirl are supposed to be smart, so it makes zero sense for them to deliberately make fighting harder for themselves. The best fighters in the world (which these two supposedly are counted among) do not create vulnerabilities for themselves because you can always screw up no matter how good you are. And even if by some miracle, no bad guy in comic book history thought to grab their hair, it would still end up in their eyes and obscure their vision due to all the movement they make in battle. The only possible excuse is if Batgirl did this at the very start because no one told her better, but Batman should have pointed out the problem years ago.
Let's avoid bringing Batgirl into this as it's not her page, just for ease of discussion (plus you could make the same argument for literally any superhero with long hair who doesn't tie it back). Batwoman's wig is detachable just like Batman's cape is. In Detective Comics Alice tries grabbing her hair but only ends up tearing the wig off, and in the ongoing (IIRC, I don't have the issue on hand right now) she gets hit hard enough that her entire mask flies off. As for the hair getting in her eyes, Kate was trained to fight while blindfolded and while exposed to various incapacitating agents like tear gas and flashbangs, so it's not unreasonable that a bit of hair getting in her face wouldn't bother her too much. I'll admit that those aren't 100% satisfying explanations, but this may be a matter of Rule of Cool or other Acceptable Breaks from Reality that you just might not ever accept, and that's fine. Also, I'm not sure where you're getting the idea that either of them is among the best fighters in the world; they're good, but among normal humans neither of them is even in the top 20 (or possibly even top 40) best in the DC universe.
How come criminals and the media don't mistake one bat-female for another, or think all the Bat-women are the same person? So if one Bat-girl goes rogue, would cops go after Kate and Barbara? Kate didn't always have pale skin, so I'm surprised people didn't mistake her for Barbara Gordon.
The general public has gotten them confused on a few occasions. In the New 52, Batgirl gets mistaken for Batwoman a couple times, and a group of thugs once confused Huntress, who isn't even Bat-themed, with Batgirl. All that confusion could just be because there aren't photos of any of them, plus they tend to operate at night, plus they only get seen by maybe a dozen people at a time, tops. The cops and (probably) the local media are well-informed enough to realize that they're different people, though.
Speaking of wigs, why would Batwoman wear a wig that is the same color as her real hair? Wouldn't it make more sense for her to wear a blonde wig if she's trying to throw people off?
A few reasons. First, much like Kate's pale skin, her natural hair color is exaggerated in the artwork for effect; in-universe, her hair isn't dyed (it's the same shade when she's six years old), so her actual hair color is likely just a regular shade of red (i.e. orange or possibly light auburn) and not literally red as it appears on-panel. The fact that Maggie never suspects Kate or mentions that she and Batwoman have the same hair color gives weight to this. Second, a blonde wig would clash with her costume. That may sound silly, but the reason her costume was designed with a black-and-red color scheme is because red "doesn't pop during night ops," according to her father. That may not be true in Real Life, but it's the in-universe justification, which works fine. So why not a black wig? Well, that brings up the last and possibly most important reason of all: Rule of Cool.