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)