Thirteen mentions that it's been a long time since she "last bought women's clothing." Since the Doctor never regenerated into a woman until now, it would leave one wondering why the Doctor would shop for women's clothing... until one realizes that she was probably talking about her granddaughter Susan Foreman.
Not to mention the Doctor's many female companions over the years. There's also the possibility that, for whatever reason, some of the Doctor's male incarnations may have engaged in cross-dressing: remember that in "The Eaters of Light" the Twelfth Doctor mentions having been a vestal virgin second class. It may not have been Twelve, personally, who did that, but there is the implication that it would have involved cross-dressing.
Given that Leela normally wore what amounted to a leather bathing suit, it's hard to imagine that Four wouldn't have had to buy period-appropriate clothes for her when they visited Victorian London. She would've been arrested for public indecency if she'd set foot out of the TARDIS in her normal attire.
The TARDIS has huge rooms with clothes for every era and has for most of the series. We don't know if the clothes there were bought by the Doctor at some point long, long ago, or were left behind by countless Companions we never saw, or are generated by the TARDIS herself. However they got there, the Doctor wouldn't have to buy Leela any clothes specifically for her, just take her to the right wardrobe.
It's also possible, assuming the women's clothing was bought for Susan, that back in those days the TARDIS wardrobe wasn't nearly so big. After all, in "Twice Upon a Time", the Twelfth Doctor mentions that the First Doctor's TARDIS database doesn't have very much in it, so it's likely that the wardrobe is in a similar state, necessitating the Doctor to buy and/or otherwise acquire such things.
Note that the last woman the Doctor would be expected to buy clothing for, such gifts to one's wife being commonplace, wouldn't actually need him to do so. River Song got her new outfits out of a spray bottle.
Tzim-Sha says that his species have the right to hunt on Earth due to "permission", which seems to be that their pods can't land on Earth unless someone, such as Ryan, presses the "button" on the golden glowing symbol that appears beforehand. However, it's possible that those symbols manifest in front of people who are likely to press them, specifically so the hunters can do their thing. Hell, maybe that's how Rahul first encountered one of those pods seven years before.
Or they basically realized that humans can't resist a shiny button, and Ryan is just par for the course.
Or they just keep sending the "buttons" down in various locations suitable for the hunt-ritual, for hours, days, or weeks, until they luck into finding somebody who does fall for the Schmuck Bait. No reason to assume Karl was chosen before Ryan touched the thing, after all.
It is also possible that it's something that the series will come back to at some point (as is the assertion that aliens don't exist) but that's purely speculation.
The Doctor landing in the train at the exact moment it comes under attack by the data coil might seem like a Contrived Coincidence, but remember what the TARDIS told them in "The Doctor's Wife"? Even when flinging them away while exploding the TARDIS is still making sure the Doctor ends up exactly where they need to be.
Why would the Stenza, who clearly have no qualms about capturing and killing members of other species, bother asking for "permission" to hunt on Earth via the buttons? Simple: intergalactic law, or something. They can claim that they were given permission to hunt on the planet from its natives, glossing over the fact that the people who press the buttons don't know what they are or what they are technically agreeing to.
After the Doctor made his Declaration of Protection in "The Christmas Invasion", it's been shown in several episodes that other races have become wary of Earth, and there are even treaties on the subject.
If the Stenza have a body temperature that's cold enough to freeze-burn human tissue at a touch, then hunting on a world like Earth is probably a bit like a human going on safari in Hell. Small wonder that they consider it an impressive accomplishment, despite humans' relative weakness ... especially if Tzim-Sha was supposed to have undertaken the hunt without his refrigerated suit, too.
The Doctor takes advantage of Yaz's fears that her superiors on the police force probably won't believe her if she tries to tell them what just happened, talking her out of calling them. However, we are talking about a reality that's had several very public alien invasion attempts, not to mention the fact that U.N.I.T. probably has put the word out to every law enforcement body on the planet to contact them if someone calling themselves "The Doctor" shows up, especially in association with anything unusual happening. The Doctor may really have been discouraging Yaz from calling her superiors because she didn't want Kate Stewart taking over (or worse, putting the Doctor in charge of the planet again).note The rest of us, on the other hand, would love for Kate Stewart to put in an appearance....
As was pointed out in the main article, this is the third episode/adventure to take place within, at most, a day for the Doctor, and it was a day for the ages for our hero. The first adventure was both emotionally and physically traumatizing — Twelve literally died believing his oldest friend Missy had betrayed him after all his efforts to redeem her, and that he'd failed to save Bill from being a Cyberman. The second saw the Doctor putting off their regeneration, dealing with the embarrassment of their less-enlightened younger self, then having their memories of Clara return, before finally going through with said regeneration. That led to getting tossed out of the TARDIS in mid-air straight into a third adventure, a rough slog through regeneration sickness while underequipped, making new friends despite herself and then immediately seeing one of them die trying to be a hero. (Another one for Davros's montage.) And yet while the Doctor declares she and Yaz are friends now, etc., like it's a typical Companion recruitment, at the end of the episode, she's sad but content to leave everyone behind. Because after all that, you'd probably need some time to process all that, and for better or for worse, the Doctor always tends to process traumatic shit by wandering around alone. That's not how things played out, but that is what she was originally aiming for.
Tzim-Sha's ritual hunt is mostly played out after dark, with him remaining hidden in his pointy-topped capsule until nightfall. This isn't just a way for him to escape notice by human witnesses, but also to avoid the warmer hours of daylight: a period when his refrigerated armor might not suffice to ward off heat stroke on a planet that, to a Stenza, is dangerously hot.
The Doctor surviving such an enormous fall, seemingly without a scratch, makes little sense at first. However, back in Tennant's era, there are two examples of him falling from a great height while remaining at least functional, and while in the former he was cushioned by an air pocket, he fell hard enough to break the visor of his helmet and he was able to get to his feet and walk around without much time to recouperate. Then consider that Tennant's Doctor had an entire hand cut off, but due to being within the first 15 hours of regeneration was able to grow it back. While the drama of the Doctor falling helplessly to the ground from a great distance is totally removed, there is at least a precedent for her remarkable recovery, especially considering just how fresh her regeneration was at the time of the fall.
Rahul's obsession with his sister Asha's disappearance and recklessness about confronting Tzim-Sha, even after seeing how destructive the alien hunter's emergence from the transport-pod had been, seems extreme if not suicidal. But that's quite possibly the literal truth, as he seems to know what the pod is even before it opens... which would only be plausible if he's the one who triggered the last pod's arrival seven years ago. He's putting himself in such danger because he believes Asha's abduction is all his fault.
Rahul's sister Asha is still trapped in stasis on the Stenza homeworld, along with who knows how many others.
Do they take a tooth from these victories? And if so, is it while they're in stasis...or before?
There are a lot of teeth embedded in Tzim-Sha's face... and he only takes one tooth from each victim. They all look like human teeth, too.
In "Heaven Sent", the Doctor states that he was transported by "an augmented ultra-long-range teleport, so I'm not more than a single light-year from where I was." Tzim-Sha's teleporter is described as "short-range". That means that it probably isn't capable of taking him safely anywhere off-planet. Tzim-Sha's capsule had come over five thousand galaxies. Either he's still on Earth, looking for escape and/or revenge, or he was teleported to the maximum range in the direction of home and then ended up floating in deep space. And given that the Doctor was adapting the same technology to transport her to her TARDIS...
Technology in Doctor Who is notoriously inconsistent, so there's always that. However, the Doctor gave Tim back his recall switch, which was linked to the pod he came to Earth in, unrelated to the teleport he uses to escape the rooftop. Presumably, that's intergalactic graded.
Of course, given that he had a quintuple dose of DNA-mangling Explosive Leash go off inside him shortly before he teleported out, whatever did wind up at the teleport's destination, near or far, may have been little more than sludge.
Karl works for his father's company in a position guaranteed to exacerbate his fear of heights. Did his father make him work in that position?