A freshly regenerated Ten originally lost his fapping hand in a sword fight against the Sycorax leader during The Christmas Invasion but luckily still had enough regeneration energy to sprout a new one.
Meanwhile Captain Jack scoured the Earth to find the Doctor's severed hand so he could jerk himself off with it. He eventually located it but before he could have a smash, Donna farted on it and the hand grew into Handy, a human version of the Tenth Doctor.
Enslavement, mental collapse and murders
After being given to Rose as a present by the Tenth Doctor, Handy struggled over the years with being enslaved, being "born out of war" and being an immortal trapped in the body of a human. He eventually went insane, murdered Rose (and Jackie) and became the Valeyard. He then went back in time to steal regenerations off the Sixth Doctor, because the Sixth Doctor was shit.
This is all completely canon.
The new TARDIS
There exists an extra bit of dialogue on Bad Wolf Bay where the Doctor hands his clone a coral-like piece of the TARDIS, telling him to grow his own. When the clone Doctor protests that it takes thousands of years to grow a TARDIS, DoctorDonna provides him with a faster solution so that Rose and the cloned Doctor can travel through space "as it should be".
This scene was deleted before broadcast. Why? Well, given the choice between an ending that's about worlds where the sky is burning, the seas sleep, and the rivers dream, or another ending that's about whether some guy will finally tell his teenage girlfriend that he loves her, RTD knows which one is truly Doctor Who.
However, the scene was mentioned in The Doctor's Data section of the Doctor Who Adventures magazine and in the 398th edition of Doctor Who Magazine. Russell T. Davies states that it is perfectly fine in his opinion to assume that this part of the scene did actually occur. The scene is included on the Series 4 DVD Box Set.
Russell T. Davies confirmed a shit.