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They've figured out some things about time travel, especially since his old partner, Captain John, came by in the Series 2 opener, and they know that he has a child (that he had borne).
Hmm. You referring to Grey? Grey's His brother. When Jack's Planet was attacked, he and his brother ran. At some point Jack lost Grey, and never saw him again. At the end of Ep1-Season2, Cap John tells Jack he's found Grey, and then vanishes. That being said Cap John is back for the season/series 2 finale, and lets just say he's not alone ( at least I don't think he is, judging from the "next week on TW" bit)
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the time travel functionality is broken on it however, but the Doctor does repair it for a while in the episode "The Last of the Time Lords" no idea at all if it is still working after thats sorted out.
Nope, the Doctor disables the time travel/transportation functionality at the end of that story arc. Saying he can't let Jack be skipping around time. After a bit of banter, Jack tells the Doc and co a bit that ties up a large subplot line.