You can never be "sure" of that, any more than you can ever be 100% sure that the plant wasn't dropped there by the space-faring inhabitants of the fourth planet of Zeta Reticuli. Certainty is not a reasonable standard to apply to this, or to most other human endeavors. You will have to settle for "more likely than not", or, if you're really lucky, "much more likely than not".
2. How, specifically, is genetic relationship determined?
As I said, the relationship is determined by genomic similarity, as measured by statistical comparisons of the two strands. If you'd like to know more about the exact mathematics of the statistical tests used, I can suggest some good textbooks to get you started, and if you'd like to know more about the exact mechanics of genome extraction and sequencing, I can solicit recommendations for where you might begin to learn about such things from some folks who are qualified to make same.
3. Which labs have confirmed these results?
I think I've already addressed this as well - as this is the initial finding, it is unlikely that anyone has yet confirmed it. Someone has to be first, after all.
Some of the elbow joints have sprouted wings and are about to fly away, possibly taking my house with it. On the face of it, this seems so contrary to common sense, that I ask the expert to explain his conclusion. He responds by saying, "you have to trust the exports."
Clever. Except that, unlike plumbing, in biology, very few people outside the field have even a rudimentary understanding of the field, and therefore they have no idea at all about what is or isn't "common sense" in the field of biology. After all, if you want to learn about plumbing, you can always start banging away on your own pipes, and eventually learn something through trial and error, if nothing else. If you want to learn something about biology, it's a bit more complicated - unlike the sorts of things professional plumbers do, the sorts of things that professional biologists do are simply not accessible to most people. I bet you have a pipe wrench in your house somewhere - have you also got a 16-capillary Applied Biosystems ABI Prism 3100 handy? Do you even know what a 16-capillary Applied Biosystems ABI Prism 3100 is, or what it does? Are you sure that plumbing is a good analogy for whether or not we have some inkling of what constitutes common sense in biology?
I think not. At best, we have some rumbling in our bellies that lets us know that this sort of thing is uncomfortable to us, but that's hardly a basis for critique. You give me, in a nutshell, "it doesn't feel right", and ask me to substitute your judgement for the plumber's, and to do it based on nothing more than that. I should think you would not be surprised if I were loathe to proceed on those grounds.
The question of evolution touches upon our existence, our relationship to God, and our understanding of truths that have sustained civilization for much longer than the tawdry reign of Charles Darwin. These are questions you simply can't leave to experts, and particularly to scientists, because--as C.S. Lewis intimated--they really haven't been taught how to think. They measure and extrapolate and boil water and drop grids over dusty chimp graves in Africa, but they simply can't be the last word on the origin of man.
Reality is what it is, regardless of how we feel about it. It's rather implacable, in that respect. And the reality is that evolution via natural selection is, far and away, the best explanation for the diversity of life on earth, regardless of whether the biologists persuade every one of the thinkers, or none of them at all. That is, either way, of no consequence whatsoever to the truth of the thing. Persuading others is always a nice thing to do, but failing to do so hardly makes it false.
And as to whether it shakes to the core our relationship with God...the philosophical and theological implications of a heliocentric solar system didn't change the reality of it, any more than the perceived implications of evolution change the reality of it. I am sorry to be the one to break it to you, but if our relationship with God is somehow contingent upon evolution being false, we will simply have to rethink our relationship with God, and come to a better understanding of it based on what we know to be true and what we know to be false. The fault is neither Darwin's, nor the biologist's, nor God's, in that case - the fault for an imperfect relationship with God, based on something once thought to be false, but now known to be true, lies solely with us. And it will be up to us to fix it within ourselves...