Of course Yukio didn't forsee Yashida's death, because he wasn't really dead.
To be fair, she didn't foresee him actually dying at the end of the movie either.
You know what, we never see his dead body after he gets thrown out the window by Logan. Maybe he's still alive?
Yashida keeps calling Logan "kuzuri" at his sickbed before his fake death. While "kuzuri" is allusion to a mythological japanese creature with huge claws, it also sounds a lot like "kusuri" or medicine. Guess "who" is the medicine that Yashida plans to obtain to become immortal.
You know how Logan acted like he didn't remember anything about Yashida and never went back to get that sword? Well that's because he didn't remember. He had bone claws in the flashback segments, meaning he hadn't lost his memories due to the adamantium bullets yet. Though there's still the Fridge Logic of why he never went back to get it after leaving his squad...
Hey this is Wolverine, who is pretty much unkillable. In his mind he has all the time in the world, besides its not like he uses it, when he has 6 blades that he can pop out at will.
While the movie seems to go out of its way not to reference X-Men Origins: Wolverine, they share a plot point that was unremarked upon in the previous movie — that super-heating adamantium lets you cut through it. Recall during the fight with Weapon XI, when is Logan able to actually land the killing blow? After blocking the eyebeams for a solid minute, leaving his claws glowing red hot.
Throughout the film, we are shown that Logan is severely wounded by cuts, gunshots, etc. Yet he can retract his claws without any problems coming from the cuts caused when the claws extend from his knuckles.