As I read your post, it is these words you wrote that stood out: "I was raised with a strict Father. I was taught you don't allow children OR teenagers to "back talk" adults; especially their parents. My wifes kids have always back talked her & this has always drove me crazy!"
It may well be that your wife's children talking back to her "push your emotional buttons" precisely because you were not allowed to talk back (have a voice) when growing in your own childhood home. Ironically, the patterns we demonstrate in our adult life were begun in our own childhood.