There are a lot of different things you can get when you see something like this. For a very new part like this I would assume that since Intel is pretty much the only company with the ability to actually build these CPUs (they use a very small manufacturing process), these are either bricks of lead attached to the correct packaging to look like a CPU, or they're failed CPUs. There's actually a relatively low yield on tiny manufacturing processes like Intel's current generation (22nm is the current size they use). I've been led to believe it's something like a 60% yield (i.e. they produce 100 processors they only get 60 that actually work) and the rest have to be discarded. But I have no real numbers on that, but even if it was a 99.9% yield, that would still mean that 1 in 1000 was bad and had to be disposed of, and Intel produces a lot of processors. And someone is probably interested in cheap, mostly functional CPUs.
What functionality is actually missing in these discarded chips could be very minor. Something like "dividing anything by 3879 never gives the correct answer", but clearly a chip with a flaw like that could never be released without permanently damaging the companies' reputation. So if these $20 i7 chips do function in a core i7 motherboard, I would assume that you would find that each one would have some subset of functionality that misbehaved. Alternatively they could only work if underclocked, or if they were much cooler than the specifications normally allow. Who knows! It's a lottery of functionality.
Another unlikely possibility is that these are some other chips which have the same pinout but do something totally different. See Sparkfun's adventure in counterfeit ATMegas (note that the post I linked has several updates where they learn it's an ON semiconductor part from the 1980s). This is extremely unlikely though, as Intel varies its pinouts frequently, so other manufacturers wouldn't be producing parts which would fit in this generations' sockets.
Something which sometimes happens in China, is that the employees will come into the facilities and run the factory when the managers aren't there, and sell the output as genuine product, even though it hasn't been tested. In general the test equipment and the equipment which marks the packaging with "Genuine __ part!" don't work without the manager's password or something similar. This is normally an issue in places that produce SD cards and similar though, where the process is relatively simple and short. These are called "ghost shift" components because they're produced by a shift of workers who aren't supposed to be there. Intel's chips are probably a bit too complex for something like that, and I think they're mostly produced in America anyway.
So long story short: I don't think there's another entity with the ability to produce 22nm parts as complex as Intel's CPUs right now, so I imagine these are either defective core i7s, or completely fake.
Edit: Or, as Olin noted below, the least interesting answer electronically: they're just stolen chips (I like my schemes to be more elaborate!)
Of course they can be soldered. That's how they are mounted to the board during manufacturing. The BGA (ball grid array) style of package you are referring to needs special equipment to solder, but absolutely it can be soldered. That's how it's intended to be used.
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Certainly. There is a lot of stuff crammed on that board, though. It's probably at least a 6 layer board. It would be a complex board to design and a bit costly to produce, but it is certainly possible. It would be difficult to get the cost down to the same level as the Edison board itself.