Karma, Axiom Of Causality & Reincarnation "everything has a cause"
Ok, but what is a "thing"? Many of these people assumed they know what they are talking about, but they didn't. Only Kant understood that there is a difference between our own faculty of mind that sees "things", and something else, beyond our "idea", he called thing in itself, and is unknowable. Of course, this sounds unnecessary complicated for simple people. But this is the only way to explain how a world of causality can ever be, what started it, if you need a preceding action to infinitum. Yes, we have big bang theory, but what was before that? Nothingness? Void? Well, it's something beyond our capacity for reason, but it doesn't mean there was nothing in existence. Existence can't have a beginning, and obviously even in the void, something existed, without need for causality. We can only imagine why it exploded and created this universe. But one thing is for sure - there was no cause, since it preceded any cause.
With that said, Karma is very overrated concept in the eastern philosophies, since it deals with physicality and the illusion of life they claim to dispel. But like anything turned into a mass religion, it needs some "rules", some sort of "moral ground" so the society can function. However logically speaking, Karma itself affects your current life, and is destroyed after the destruction of the corporeal body. However, what's left is the mental direction, which has nothing to do with morals or judgment. And actually this was proven by the few real researchers that investigated children memories of their past life. IMO, there are 3 very different layers we exist in. One is this Karma/Causality - I call it the dumb level. It can be explained with these simple rules you said, "Eye for an eye", right. Higher is pure mental level, which can be very complicated. It's like "turn the other cheek". Many do not get this idea, but in it's core this is just to reverse the rule and think. The real mental level is vast, causality is like... 0.000001% of the possibilities. And even beyond that is yet another level, we can't even comprehend. It's the void, or pure being / nothing as Hegel calls it. Nothing means just - without any concept about "things".