Jesus' DNA would have to have had a component other than Mary's, I believe, as her egg would have to have been fertilized somehow. According to orthodox Christian belief, that part would have come from the Holy Spirit.

As far as I know, the Catholic Church has never claimed to have a sample of Jesus' blood. If you are referring to the Shroud of Turin, I believe it was carbon-dated to the 14th or 15th century. I would call it a work of art rather than a 'fake'. Pope Paul VI once said, I believe, that he believed it to be genuine, but stressed that it was only his personal opinion. That was, of course, before the dating procedure.