Why does Reyes come off as so bitter in her letter to Roth about how he cuts Lara too much slack, reminding him harshly that Lara isn't actually his daughter? Because Roth has a daughter, even if Reyes never told him that Alisha is his. She was likely even harsher than she would have been previously, since she was considering finally telling Roth about Alisha.
Also works as an explanation for why she's generally the most hostile towards Lara amongst the Endurance crew throughout the game.
Why can't Lara use her climbing axe in combat until the second upgrade tier? Because it's an incredibly brutal method of killing someone, and Lara can't handle that at first. The game even calls her "hardened" at that point.
As noted on the main page, the actions taken by Himiko in the final act only serve to aid Lara in her ultimate destruction of Himiko's body. Why would she interfere when she literally had to do nothing in order for Lara to fail? Because she wanted Lara to succeed in destroying her body and setting her soul free. Up until Lara's arrival on the island the only option for Himiko was to coerce Mathias into finding her a new host. When Lara showed up and learned exactly what happened, why Himiko was so pissed off and how to set her soul free she allowed Lara to climb the monastery in order for her to destroy her old body and set her soul free, once and for all.
Except by that logic, it would have been easier to have Mathias kill Himiko instead of doing the whole host-body thing. Besides, it was heavily implied that Himiko was evil, and would have preferred to get a new body and Take Over the World, rather than simply die and most likely go to whatever form of Hell the Yamatai people believed in.
Himiko can't directly tell people what to do, otherwise she could have told the Stormguard to stand down when Mathias showed up with Sam. Best she can do is whisper to people in their sleep and that's assuming Mathias isn't just hallucinating that. As for the evil, taking over the world part: She just spent at least 1,500 years trapped in a dead body. That's plenty of time of have a long hard think. In the end, she hedged her bets; new host or soul set free, either option is better than her current state.
After climbing the radio tower, Lara successfully manages to hail a rescue plane. This being relatively early in the game, I suppose we can all guess how it's going to go down. But even if the plane did not crash, where exactly was it supposed to land? There is nothing on the island that even closely resembles an airfield. And even if smaller aircraft can technically land anywhere with enough open space, this particular one is a rather sizable transport plane which is very much going to require a proper runway.
Presumably there's an airfield dating to World War II on one of the parts of the island Lara doesn't travel to.
It doesn't have to land. Once it's spotted the survivors, it can note the location of the island and a full rescue party can be send out by ship.
Okay, late in the game, Mathias sends Dr. Whitman to let the Oni/Stormguard know that they have a suitable host for Himiko. It seems that the Oni don't understand what Whitman is saying, but apparently they're willing to hear him out for a minute or two before killing him. After speaking to them in English the whole time, Dr. Whitman tries saying the Japanese word for "Queen". How do the Stormguard respond to this most joyous of news? They get angry and immediately kill Dr. Whitman right there on the spot. Am I missing something here?
His mangled speech and weird behaviour makes them think his use of the word "Queen" is threatening her.
Why are there parts for a Desert Eagle hidden away in ancient tombs?
Perhaps those parts were left there by the Solarii, who have discovered quite a few tombs that Lara visits. If they already believed in Himiko's myth at that point, it is possible that instead of robbing the tombs blind, they added something valuable of themselves to the treasure chest as a gift/offer for the Sun Queen. On an island such as Yamatai, where the strongest rule, guns and other weaponry were probably considered one of the most valuable things one could possess and thus made for a suitable offer.
Lara's using wood and animal pelts to add alternate burst modes, silencers and larger clips. Using a Ruby to magically turn a Type 100 into an AK-47 is child's play.
If you look carefully in the shipwreck area, Lara finds some children's toys and their father's wallet lying around. Lara herself even asks herself what must have happened to them. Given how brutal the Solarii Brotherhood are, it's not hard to imagine this innocent family must have suffered a very gruesome fate.