In the quest "A Dog Astray", you are asked to go find an old man's dog. When you find it, it has already died, and the old man has passed away by the time you return to him.
Although the reality of the overarching series is indeed unstable, this particular game does not delve into that topic. There are some brief hints left for people who have played the other games or read the supplemental materials, but it would be difficult to catch those hints unless you already knew what to look for.
It’s not a hospital, but one of the dungeons is inside a research facility. Human experimentation is mentioned in text form and at least two subjects are child-coded.
Not present but briefly discussed. There's a scene where a young woman shows up with bloodstained clothing, and a minor character assumes she is menstruating. He is mistaken, but you don't find out about that until later.
Kaine, who is implied in the game and confirmed by the director in an interview, to be intersex/a hermaphrodite. At the end of the game (during ending C/D) you are given the choice to kill her before she dies an agonising death from Shade possession or sacrifice yourself to give her a Replicant body. Sacrificing yourself is considered the canon ending however.