I'm having trouble seeing what evolution has to do with it. Many animals, Singer supposes, feel pain. Pain (roughly; the refinements won't matter here) is intrinsically bad, no matter what sort of creature experiences it. Whether animals (or humans) feel pain because of evolution, because a God made them that way, or because we're all sentient animaldroids, designed by mad scientists from Mars is beside the point. Singer's thought is that pain is bad for us, and animals are no different from us in that respect. He isn't making a point about the phylogenetic tree. Cabbages don't feel pain; cats do. So when we're calculating the balance of pleasure to pain, the cat's pain (or pleasure) should be included in the calculation. But since cabbages aren't sentient (so far as we know), there's nothing about the cabbage to add or subtract.
Part of the problem is to decide what counts as "tampering with the natural order." In at least some senses, we "tamper with the natural order" all the time. Modern medicine is a clear example, but you could even make the case that selective breeding of the sort that farmers and gardeners have practiced for centuries is another case. Most of us don't see these as wrong.
It may be useful to step back and look at the phrase "playing God." If there is a God, and if that God has designed a providential plan that works to our benefit and if some sort of intervention would amount to thwarting that plan, then that would be a reason for not making the intervention. Those, needless to say, are big "ifs." However, even if we grant them, we're left with the problem of deciding which sorts of interventions would count. God's plan -- even if there is one -- isn't as clear as some would like to claim.
But let's leave the theological issue aside. You ask whether tampering with the natural order is acceptable if we understand the consequences and know what we're doing. Let's grant for argument's sake that it is. The moral that some "don't play God" stories suggest is that these "ifs" about consequences are also very big. The worry is that some interventions might have large, unintended and undesired effects. On that reading, such stories are cautionary tales about arrogance and lack of due regard for what we don't know.
It's hard to deny that there's some good sense here. Large, sudden disruptions of complicated systems often produce unintended consequences. And indeed, in some cases one might reliably predict that our predictions will be unreliable. Stripped of debatable theological overtones, there's something to the worry about "playing God." The practical problem, of course, is to sort out the cases. Not all interventions are as risky as some would claim, and sometimes taking chances pays big dividends. But turning those platitudes into detailed advice would take this philosopher, at least, well beyond the realm of what he knows.
If "natural" means "part of nature broadly conceived," then it's hard to see what's uncontroversially not natural. But what this really shows is that there is mre than one meaning of "natural" and more than one contrast that someone might make.
Someone who believes that the material world was made by an immaterial creator would contrast the natural with the supernatural. On that usage, more or less everything in space and time would count as natural.
But someone might also have the distinction between natural and artifactual in mind, and if that's what they mean, then my computer is not natural, but the flower on my windowsill is. No contradiction here; just a different distinction. As for what makes something an artifact, that's not easy to say with real precision. But it's easy to come up with a wealth of examples that more or less everyone will agree to. (The fact that we can't articulate a distinction doesn't show that we can't make a distinction.)
There's another notion of "natural" that's broadly teleological. It makes most sense, perhaps, if we assume that there is a designer who made the world according to a providential plan. On that way of lookng at things, we and various other parts of the world were endowed with natures that, when working properly, contribute to the providential order. On such "natural law" views, what distinguishes humans from animals is that we can choose to act contrary to our natures. Thus, a natural law theorist might say that we were endowed with reason, and that to ignore our reason is to act contrary to our providentially-ordered nature.
There are non-theological analogues of this distinction that appeal to something like biological function. Unfortunately, the moral weight of theological notions of divine order sometimes gets carried over into the non-theological context, and leads people to say that, for example, homosexuality is wrong simply because homosexual sex is "unnatural" by virtue of not serving reproduction.
Someone might also say that a biological process gone awry is not natural. And in fact, it's interesting to note that we can all think of examples of what we'd count as "biological processes gone awry." Whatever weight we put on the associated notion of "natural," it doesn't seem merely to be empty.
So there's no one thing people mean when they use the word "natural." And so whether something counts as natural will depend, among other things, on what notion of "natural" is at issue.