With regards to your first two questions, by all means mistrust the government to the highest level convenient. I can answer (mostly) your final question.
As of right now, with our current level of genetic analysis technology, it is not possible to determine anything useful from the tests employed in these situations aside from the degree of likelihood that it matches another sample. This isn't a safeguard set in place against the fears you named; it's a coincidence coming from the tools we have with regards to cost, ease of use, and most importantly, speed. What these tests try to find is short, predictable lengths of DNA through a variety of different methods--it's by no means a full DNA sequence. Furthermore, the location of these lengths of DNA isn't determined, only their presence and how many of them are found in a single sample, so it would be impossible to determine if they were part of a known gene.
That's as of now. Our ability to fully sequence DNA samples gets better yearly.