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.
The entire "Experience" mechanic is integrated into the game quite nearly. Every skill she learns is something that she could either learn on the island, or a brutal attack she'd become more willing to do after fighting for her life for days.
Learning to scavenge items and ammo better and how to carry more on her person. Be more willing and capable at scavenging human bodies. She gets an eye for spotting rare items using her survival instinct (ability) and marks them on her map.
Not only becomes willing to wield the axe in combat, but becoming more effective and brutal with it.
Being willing to fight dirty and brutally. Throwing dirt in her enemies' eyes, dodge and then stab them in the knee while they're off balance. Bonus points for her learning how to do this lethally (stabbing them through the neck) and then use it against more elite troops.
The final unlocked tier has her full out execute injured enemies in a ruthless fashion with her weapons, either with point blank shots to the head (with her pistol and shotgun) or knocking them down and cutting loose on them (with her assault rifle). Even slaughtering them by using her arrows as melee weapons if you have the bow equipped. By the time you unlock this tier, Lara's in full "The Dreaded" mode and taunting her enemies.
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. In fact, she was likely hedging her bets: If Mathias wins, she's free and in a fresh new body. If Lara wins, she's finally dead, her torment over. This is supported by the fact that the Stormguard who kill Whitman, give chase to Sam and Mathias but her blown away by Himiko's galewinds, but only when Sam is already safe off the bridge (which flies off along with the Samurai).
A more likely explanation is that Himiko doesn't have fine control over the storms. She can direct them in a general area (e.g. to take down a plane or ship), but it's unlikely she can control where lightning hits or how a piece of wood blown by the wind will move. It's mentioned that the storms are due to her "rage" after all. So Himiko was likely just directing the storm at Lara, and the results involved a large degree of chance.
Lara doing her dual-wielding is more than just a callback to her previous incarnations; By the end of the game, most have upgraded via the Magnum parts, but remember, the pistol Lara has is Roth's. Roth wielded two pistols; one Lara kept on her, the other she gave to Sam, which was taken by Mathias. Not only does Lara defeat the enemy this way, but she reunites Roth's dual pistols and, in effect, balances herself, being off balance for most of the game.
Lara's Survival Instinct is pretty realistic when it's not upgraded. It's basically combining the senses of smell, sight, and hearing to augment her awareness. She might not be able to see an enemy with her eyes, but she can smell or hear them well enough to know the general area they're in, which translates as us being able to see them via highlights. But it's incomplete, as can be seen when it cancels out every time she starts walking around while it's activated. She's inexperienced, so it's probably taking almost all of her focus to use her senses like that, so moving around too much would make her lose said focus. Chances are Roth has a better version of it, but only due to experience, but it is also imperfect, most likely due to age making his senses decline, but if he were in his prime, or once Lara gains enough experience, moving while in this state could definitely be possible.
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. The father can actually be discovered as one of the insane prisoners trapped in the Sulphur Mines. Apparently, the Solarii took his two little girls and murdered them right in front of him.
Himiko is later revealed to be have been trapped in a rotting corpse after Hoshi committed suicide whilst Himiko tried to possess her. It's never made clear whether or not Hoshi isn't trapped along with her. This can potentially explain why some of the storm effects seemed to have been helping Lara.
One has to wonder about what happens after the end of the game, after Mathias is killed. Do the Solarii know that they are now able to leave the island? Considering how slowly news can be seen traveling by listening to enemy chatter, chances are a lot of, if not all of them will stay stranded on the island just because of this fact. If that isn't enough, think about the "unholy ones." Most of them are caged up, but they can get loose. Lara canonically probably didn't kill them. They'll probably stay there, escape, breed, and form a whole civilization of cannibalistic humans on Yamatai.
Fortunately, according to dialog in the sequel, Trinity exterminated them.