The Doctors Pockets are only Bigger on the Inside for the doctor.That's why the hospital staff wasn't pulling mountains of things out of his jacket.
Grace and Lee regenerated in the Cloister Room.It's been established since the end of Doctor Who S11 E5 "Planet of the Spiders" that regeneration energy can be influenced or even transferred from one Time Lord to another. And it can be inferred that, in trying to restart his life cycle, that the Master took a considerable amount of that energy from the Doctor and absorbed it into himself. So, when the Master's body was consumed by the Eye of Harmony, some of that energy was released and, directed by the TARDIS, was sent to Grace and Lee, allowing them to recover from their recent deaths and return to life.One factor supporting this would come along in 2013 in Doctor Who 2013 CS "The Time of the Doctor". When the Time Lords give the Doctor the regeneration energy to restart his own life cycle, the visual effect looks a lot like the energy released from the Eye in the earlier movie. Yet another way that the 1996 movie would influence the later revival.
The Doctor may not have been Genghis Khan, But the Master was at some point.At one point, the Master tells Chang that the Doctor was Genghis Khan, we know this can't be true as all the doctor's previous incarnations up to this point are accounted for. However, most of the lies that the Master tells Chang about the Doctor are actually truths about himself, maybe this was no exception. It is entirely possible as we know that Roger Delgado's Master was the thirteenth Master, that leaves almost an entire regeneration cycle's worth of incarnations unknown, making it entirely feasible for the Master to have been Genghis Khan at some point. This lends itself to a bit of Fridge Brilliance in "Rose", when the Ninth Doctor claims that the assembled hordes of Genghis Khan tried to get into his TARDIS, why would Genghis have been chasing the Doctor? because he was actually the Master!
The "half-human" comment is a glitch foreshadowing the Last Great Time War.One of the ripple effects before all hell breaks loose is the Time War messing with the Doctor's history, briefly causing him to be Half-Human Hybrid despite never mentioned before and ignored afterwards. The various canons surrounding the Eight Doctor are more serious results of the Time War making an utter mess of the timeline, since he was the one who entered it in the first place, with this belief being an early warning sign.
The reason why the Doctor forgot to check where he landed...Is because of the absurdness that we know as Doctor Who 30th AS "Dimensions in Time". The events that took place did happen, but not necessarily the way how it was broadcast to us on the other side of the fourth wall. Since nothing really made sense due to all of time breaking down in Doctor Who 30th AS "Dimensions in Time" the Doctor suffered mentally, causing him to make some minor errors such as not knowing how to properly operate a sonic screwdriver or not checking where the TARDIS landed.