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No, "The Texas Chainsaw Massacre: The Beginning" (2006) does not feature the Hitchhiker Girl (played by Lauren German), who appeared in "The Texas Chainsaw Massacre" (2003). In "The Texas Chainsaw Massacre", the Hitchhiker Girl was picked up by Erin and her friends before they encountered Thomas Hewitt and his psychotic family. The Hitchhiker girl has previously had a run in with the Hewitt family, where her entire family had been killed. She subsequently committed suicide. The Hitchhiker's storyline would take place in 1973, along with the rest of the events of the 2003 film, while the prequel takes place four years earlier in 1969. So there was a large gap of time between the events depicted in the prequel and the 2003 film.
Jedidiah (played by David Dorfman in "The Texas Chainsaw Massacre") and Henrietta (played by Heather Kafka in "The Texas Chainsaw Massacre") do not appear in "The Texas Chainsaw Massacre: The Beginning". The reasons are never specified, but it is alluded to that they simply are not a part of the Hewitt clan yet, or they were in a league of absence. In the fan script "The Texas Chainsaw Massacre: The Legacy", the story reveals that Henrietta attempted to escape the dying town and traveled to Los Angeles, where she became addicted to various drugs and contracted a rare disease. Jedidiah is her son in the story who travels back with her to the Hewitt's when she needs care and ran out of money. She hitchhikes home, to where she is picked up by six boys and she ultimately leads them to the clutches of the Hewitt's and Leatherface.
Eric is the first to go. He gets chained to a table and brutally sawed open (Thomas later cuts off his face and sews it to his own). Bailey gets her throat slit. Dean dies in an attempt to save Chrissie; Thomas comes up behind him and shoves the chainsaw through him. Chrissie is killed when Thomas pops up from the backseat of the car she's driving away in and chainsaws her through the seat. So, no one survived.
No. The song Alright Now did not come out until one year later, 1970. This is a mistake made by the director, because they played this particular song at the beginning of the film, when Dean and and his girlfriend were in the motel room.
Even though the Unrated version features the "Unrated" label there aren't that many new scenes containing violence. Mostly, one can find extended story sequences or lines of dialogue that have been added for the Unrated release. A detailed comparison between both versions with pictures can be found here.
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