Ok, I've read all 9 pages of this thread...I'm so glad I found it! I've got a lot to say so I apologize in advance for its length
My DD is only 3 months but when DH and I are out with her (he's white, i'm black), and people peak in the stroller, then glance back at us, you can see the wheels turning in their head as to the baby's make-up. But so far, we haven't had to answer any questions from strangers, we've only gotten compliments as to how beautiful she is, and there's no arguing that
We have, however, received a fair share of questions & statements from my MIL which probably makes up for it--"When is she going to get darker/Her hair going to turn curly", "Are you part Hawaiian cause her eyes look almond shaped", and of course the dreaded "She's so EXOTIC" [public service announcement: limit your use of exotic to objects, not people], but the MIL is a subject for another thread/another time...
I can relate to so much of what has been said. Asking someone "What are you" may not seem like a loaded question to some, but is very offputting to the person on the receiving end. Because I am fairly light-skinned, I've gotten that question most of my life, and honestly, after 30 years of hearing it, I'm over it. Over the "oh but you're pretty for a black girl", "you're so lucky to have good hair", "oh you're so light"--said in a tone of disgust, or even the assumption that because I am on the lighter side, I've somehow had an easier life or that I couldn't possibly be black. And of course, the racial guessing game..."Are you egyptian/is your father or mother white/where are your people from"...Sigh. (I'd rant more, but the "what are you" thread is closed).
To me, "What are you" has always felt like a very abrasive question under the assumption that since I'm not a cultural "norm", I shouldn't mind that not only have I been deemed different in a country that calls itself a melting pot, I should accept that my background be subjected to public dissection.
Don't get me wrong, I love discussing cultural and different ethnic backgrounds, and there should be an open dialogue for this sort of thing if the recipient is open to it, but if you're going to ask someone a question about their personal background, how about approaching it with care and a bit more respect? How about instead of "What are you", asking: "Do you mind If I ask what ethnicity you are?" Ask me that way, and there's a good chance that I'd tell you that my on my father's side my G-dad was west indian and my G-mom was creole...and so on...
The same can be said for addressing a child's background. When I hear "what are they", it sounds more like you're expecting the outcome of some weird science experiment. Asking "are they yours" makes me think more of things than people, like a new car or a shoe collection. It objectifies them, but that's just my humble opinion. Take that with a grain of salt.
Even if you aren't sure the child and the guardian with them are "together", instead of asking "Are those children yours", what's the harm in saying "Aww, [he/she/they] are so beautiful, do you mind if i ask what their background is?" instead of assuming that a) the adult with them isn't their primary caregiver/parent and b) especially if the child is aware of the question being asked, not making them feel uncomfortable about their own family situation? Starting with a compliment, then asking "the question" respectfully will feed your curiosity a lot faster than assuming that since a parent and child look different, they aren't together. Even if the person with them is the babysitter or nanny, there is no harm done--you haven't assigned them as the parent, and if they do know the child's background they may be open to explaining.
In the case of adoption...If you again, start with a compliment, but then ask "may I ask about your family situation?" it sounds waaay too intrusive, and honestly, no one's business to begin with. It may be best just to start and end with a compliment, and then move on about your day. If the adult with them cares to share, they will, if they don't, they won't. Don't press them to satisfy your own curiosity, especially if the children are older and within earshot of the question. They may already feel a bit uncomfortable if their adoptive family looks drastically different than they do. Getting that question may make the child feel worse about being different and maybe a bit paranoid that everyone else notes their difference as well. In that situation, put yourself in the child's shoes and think about how comfortable you'd feel hearing that you possibly don't belong. If the parent does not open up to you, what's the harm in just assuming that they are a loving family who were open enough to take someone else in and love them as their own, and feeling good that there are good people out there that can extend their homes in that way?
That's my two cents
: I've gotten that. I've also heard plenty of "Well, you might be black but you're still very pretty." Or "Thank goodness you have nice hair!" Or the best one (from guys): "I'm not usually attracted to black girls but you're different." How messed up is that?! As if I should be flattered, or something. 
Once I answered back, "Well, I'm not usually attracted to morons and you're no different." 
I kid you NOT.
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DP, DS1 (who was about 4yo at the time) and myself all thought it was pretty funny. When I was about 17, people used to ask me if my 1yo cousin (who looked so much like me) was mine. It's probably because I was very motherly towards her when she was in my care. The same thing happens to my 21yo sister when she has my kids; even sometimes when I'm THERE with her; and she doesn't look much like them at all (we're all pasty-white with sharper features, and she's got rounded features and is of tan color). What I'm trying to say is that it doesn't necessarily matter what the kids and the adult look like.
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