Morality is innate in humans, put there by natural selection.
You just reduced morality to "ways of acting that make it more likely for your genes to proliferate". Evolutionary morality isn't something we "should" do, it's simply a way of life that, on average, spreads our genes.
1.) That gives you nothing to say to anyone who simply doesn't care about the proliferation of their genes. Sure, you can point out that this kind of person tends to weed themselves out of humanity, but that's all. If you're faced with someone who prioritizes anything else for any reason--accumulating wealth, having the most possible sex, experiencing the most possible sensation of pleasure, getting away with elaborately planned-out pain & suffering for the sheer challenge of it--then you have no basis for saying that they should have any other priorities or behave in any other way. A sociopath isn't mistaken in any sense; they're just different. There are no real "moral values" beyond gene survival, and the instincts that lead to it, and there's no reason for anyone to value them.
Saying that you "should" do something because it will make your genes survive is almost as morally empty an argument as saying you "should" do something simply because otherwise God will punish you. And that's precisely what "natural selection instilled morality in us" means.
2.) You have implied, quite oddly, that the "right" thing to do will always be a survival characteristic. You've excluded the possibility of ever arguing that something is the "right" thing to do unless it is connected to gene survival. And you've implied that anything that, on average, decreases the survival of our genes is wrong.
That's the main argument I've heard about atheism & morality, growing up in evangelicalism in America: Not that atheists can't be moral, but that objective morality has no rational basis in a materialistic worldview. In other words, atheists who do believe that morality objectively matters--that it "should" be followed--are being internally inconsistent, and borrowing from a non-materialistic worldview.
Sure, I sometimes I do hear theists saying things like "Atheists can't be moral", so your "yes they can" response to that is valid. On the other hand, I've also watched atheists refuse to hear what theists were actually saying: I once watched a debate where a Christian argued very carefully that "atheistic morality is internally inconsistent", and specifically clarified that he wasn't saying that atheists can't act in moral ways for moral reasons. And the atheist rebutted with "Of course atheists can be moral!" It seemed like willful misunderstanding. (It was a few years ago, but I'm pretty sure it was either Christopher Hitchens or Sam Harris--it was during the early prominence of the "New Atheists". I don't think it was Dawkins, though.) I saw the same thing happen in an exchange in a college newspaper.
Of course, evil acts do not require religion, however religion discourages critical thinking in a way that can easily justify immoral acts.
I'll grant that much religion discourages critical thinking--but the claim that all religion does so is the result of either narrow experience or narrow-mindedness. (Or perhaps it's the result of the common misconception that "faith" means "belief without adequate justification", when it's actually a synonym for belief/willingness-to-rely-on-something.)
And I'll counter that discouraging critical thinking isn't limited to religion, either. (1) Self-described "skeptics" are often in reality sophomoric denialists. (2) The better point of pointing to Stalin & Mao's mass murder is that the tendency towards self-justifying immoral behavior is a human tendency, not a specifically religious one. Religion can be a very useful tool for that kind of self-justification, but that's not an argument against religion.
 There is also the argument that the rejection of God is itself immoral, and the argument that all of our efforts to be moral end up being tainted by selfish/immoral motivations.
 There's also the "how can you know what's moral without revelation from God?" argument, which is fairly common among Christians, but which is oddly ignorant of the Bible: In Romans 2, as part of Paul's argument that everyone can rightly be held accountable for their sin, Paul talks about how even "Gentiles" (who didn't have the Law revealed to them verbally) instinctively follow the Law. It's "written on their hearts", and we still violate our own consciences.