I don't know anything about DNA at all so my answers are probably completely wrong!
1) There are still living descendants of Mary Boleyn, including Princes William and Harry, but I'm not sure that the Royal family would offer their DNA for testing.
2) There is a locket which contains a lock of Mary Tudor, Queen of France's hair (sister of Henry VIII) so I guess that could be tested against a descendent of the Careys.
3) Although public opinion is that Anne was innocent and that she suffered an awful miscarriage of justice, she was found guilty in a court of her peers and I think it would be hard to go against that when we just don't have any evidence to definitively prove her innocence and we are judging it over 400 years later. Yes, she deserves justice but I don't think that a royal pardon will ever be given.
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IF, and that a big if, any type of DNA could be done, mitochondrial DNA may be possible because that is only passed from the mother, so itd have to be traced through the female line all the way back. it was done with the remains of tsar nicholas and his family to prove it was the royal family in the grave. its actually quite interesting!
I just want to add, it actually is possible if not, likely that the Queen would offer up Harry or Williams DNA for comparison. When they tested the remains of Tsar Nicholas II of Russia to make sure it was the Romanov’s bones they used her husbands, Prince Philip’s DNA to compare with the bones. In fact, I *think* it would be Harry or Williams choice, they may need the Queens formal approval, butI’m pretty sure Prince Philip very willingly gave it to them.