We'll see.
Is this the same Karen Starr from Mister Terrific?
Yes (the upcoming reveal comes in MT #8 according to Levitz)
As far as a killing Bats goes, I'm cool with that (he used guns in the GA and in First Wave just like the Crimson Avenger and Manhunter/Paul Kirk did during the GA... heck, all three of them debuted in Detective Comics around the same time)
Not to mention snapping necks (and possibly cashing checks), throwing guys off of buildings, and hanging giant mutants by the neck from his gyrocopter.
That is true, but remember, all of that happened in 1939-40. After that, Batman *totally* never killed (according to the writers and editors). This was not just about "thinking of the children" (though it was certainly a factor), but also to make Batman look less like The Shadow. The Golden Age extends to at least around 1950 (depending on whether you have an Atomic Age between the Golden Age and the Silver Age).
But as I previously said, let this new Earth-2 Batman have a gun, as long as Superman is not flying, and Wonder Woman is tying me... er, some evil-doer up, I mean.
There is one form of revisionism I like to see, though, and that is if we're going to have a Robin in Earth-2 (and that is a big if IMHO), give him some frakking pants. Or, I dunno, *something* to cover his legs.