The player character, who is a wolf, can get slapped around by certain character out of battle. Most notable is by Mushi's Mama who punishes anyone who messes with her turnip garden with a smack and has been implied to do it Hayabusa, Mushi's dog, when they to try just that.
Not only did an animal character die in the opening, it's also mentioned another character's old dog died protecting their owner's father. Other than that, the same character who dies in the opening show up before they got hurt due to time shenanigans and more realistic still 2d art depicts them bleeding.
To address another comment, the sidekick, named Issun, is NOT a bug. He is a tiny (one inch) person called a Poncle wearing a hat that LOOKS like a bug, but in his character portrait he is a silhouetted humanoid. All Poncles wear bug hats. There are also regular bugs.
What everyone's saying is a possession was actually a demon taking on the form of someone who died. You end up finding their corpse while the demon's still active
It's possible for the player character to drown, but you never see the character sink or anything. The player just stops moving and then is transported back to land.