Like what was mentioned with the domestic abuse question, there are technically events where two randomly selected cats can hurt each other, with odds improved by more violent skills or traits, so a parent can technically hurt their child, but it's rare
Technically, you can possibly get events where a cat could harm their mate, such as if they're ordered to do so by their leader, but these are not inherent to mates, and the traits you need from your leader to get this combined with the randomly generated nature of this game make this extremely unlikely
Animals which cats use as prey, such as birds or mice, are killed (without graphic imagery) on hunting patrols, and there are also depictions (sometimes with imagery, but there are options to tone down the level of blood/mild gore) of other animals dying in patrols
There usually isn't visual imagery of this (most of it is in text, though if having an animal die in general is an issue, that's unfortunately unavoidable in clangen), but there is one hunting patrol of a fox next to the corpse of a large animal native to your clan's region (ie. A deer for forest camps, a shark for ocean camps, etc.)
Grief events, moon events describing your cat's reaction to injuries or sickness, thoughts, some patrol events and more are all sources which imply that your cats might be sad, and these are largely unavoidable
Aside from clan members who used to be pets dying, there are also moon events where kittypets you've met are rumored to have passed on and one patrol event where an abandoned kittypet is run over
The game does not allow kits or adolescents to form romantic connections with adult cats, and has a cap on age differences in regards to romantic interactions (with the automatic restriction being that cats with an age gap of over 40 moons cannot fall in love. This can be adjusted in game_config if even this age gap makes players uncomfortable)
Your cats can experience dislocations as the moons goes on. If I recall correctly, there's also a chance that they develop a permanent condition once they recover.
There's still a status where it says that a cat almost choked on their prey, though. No other depitions than that however! It also doesn't affect how you play the game
Cats can be killed by fires raging through camp (especially in mass extinction events, which can be disabled, though it is also fully possible for cats to be burned and die from it later)
Deadnames technically aren't much of a thing due to cats keeping their names after transitioning unless you choose to change them, but as soon as you change a cats name, the game will call them solely by the new name you chose
Direct family such as parent-child, siblings, aunt-uncle, first cousins, cannot be mates. Distant family may be, but how much that upsets you is up to you to decide, as the game doesn't say they're related – you would have to track families yourself to know.