It is unfortunate that they decided to go for the TV route, rather than publishing the paper in a journal first, as it means people don't think it's "real" science. But, do you honestly think these eminent scientists, would have been happy putting their reputations on the line, on TV for the whole world to see, if they weren't sure? One of them refused to have Richard's standard laid over the box of bones they'd just dug up purely because they were uncertain at that point in time. Reputation is pretty much everything to a scientist, it governs who will work with you, who will employ you, your promotion prospects etc. Any data fraud in a case as high profile as this would result in lifetime unemployment in their chosen career.
Regarding the DNA, you are right that standard DNA testing over so many generations would have resulted in many possible ancestors. However, there are other techniques rather than the standard DNA tests you are talking about, that they probably used. I think the most likely is analysis on DNA found in mitochondria, which is passed exclusively down from mother to child. I believe they were able to match this to the last surviving descendent of Richard's mother.
Face reconstruction is more of an art than a science, and it is difficult to be objective when (famous) portraits exist, but I think it would have been revealing if they'd decided the skull could not support the face structure portrayed in his portraits.
My maiden name used to be unique on Google, and can't see anyone else in the first five pages or so, so I guess it's still true. I use it as my professional name, so linked in etc have flooded google with crap links.
My married name has a bare four pages on Google, mostly to do with running, and it looks like there may be one other living in Texas.