The Doctor being half-human would explain why he's so protective of this planet when he has the entire universe to play in.
It stumped me for a bit, how did the Master get inside the TARDIS when Lee had the key? He used the one the Doctor kept above the door in a cubbyhole above the 'P'! How he knew about it is another question... but the Doctor and the Master have known each other a very long while after all...
Alternatively, the Master picked the lock to gain entry. He was doing so in "Survival", but was interrupted by the Doctor before he could succeed. With no such distraction this time, he was able to break the lock.
While the Daleks' decision to prefix an extermination with a trial is often considered uncharacteristic, Big Finish play We Are the Daleks has the Daleks put the Doctor on trial - because they want the galaxy to know they're about to off a legendary foe. With that in mind, their trial and execution of the Master essentially boils down to having bragging rights.
