There were rumors put about at the time, certainly false but pernicious nonetheless, by both Mary's and later James's enemies that James was the son of David Riccio. Darnley himself was taunted with this by the conspirators who gained his participation in their murder of Riccio. It was a matter of great import at the time, because it was considered essential for Mary's cause that Henry publicly acknowledge that the infant James was his son, which he did. Caroline Bingham addresses this in her excellent biography of Darnley, as does Pauline Croft in her biography of James VI & I, in which she discuss the way in which it was used against James by those at Elizabeth's court who opposed his succession to the English Crown.
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Mary Stuart seems to have been infatuated with lord Darnley when she married him, but quickly discovered his less pleasing behaviour. There is no reason to doubt he fathered their son.
The pregnancy of her third marriage like the marriage has been both described as the result of Mary's extra-marital affair with Bothwell during her marriage with Darnley and as the result of being raped by Bothwell forcing her to marry the man who made her pregnant.
She miscarried the twins days before her forced abdication.
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No. It has been said that Mary Queen of Scots did miscarry twins by the Earl of Bothwell.