a dog pokémon is severely injured, but it doesn't die. in fact, you heal it throughout the game. with sandwiches!
there are however, ghost dog pokémon, so if that's triggering to you, then i recommend not playing the game
Major plot spoiler ---
One of the character talk about how their parent was never home, then after a while stopped writing. Hes clearly upset about it and feel abandoned.
Later we learn the parent died.
there are multiple pre-existing spider pokémon, such as galvantula and araquanid, as well as two new spider pokémon which are mentioned in the other comment
adding to what mazertime said:
i didn't need to go there to beat the game. in fact, i hardly had to go anywhere in the school. unless you're talking about the pokémon centers?
The legendary pokemon is afraid to fight, suggested by one of the characters it might be cause they lost one badly.
It is later discovered that they were attacked by another from their species. They look scared.
Im not sure it count as PTSD
there are however, ghost dog pokémon, so if that's triggering to you, then i recommend not playing the game