Good question... I have actually had both a pet rat and mice as a kid and I find them very cute too. But I also spent a large proportion of my childhood on the countryside where death of animals was an everyday thing. I have also had to see many of my pets go, so I don't really get anxious by seeing a dead animal. I only get anxious when I see a living animal suffer. Not that the mouse didn't, but a cat killing a mouse is simply the way of the nature...
Well I know that this is how this world works (not just animal world, human just the same - a lot of the time even worse) but I don't like it at all!!! why would I need to like it just cause "that's the way it is"?! I don't like! I hate it! I hate that living beings are hurting eachother!!! Don't care if that's the way it is and if that's the way it always has been and if I can't do anything about it, I don't care why or how it is this way or who or what created it... I do not like it!!!!!!! I hate it!!!! OK? living beings should be helping eachother!!!! In my world! and should feel what other living beings are feeling! and should respect every life! that's just the way I see it!
That's why even though i love countryside I would never ever want to live on a farm with animals, cause it would just kill me to know that animals that I love are getting killed!!!!! I would not survive that!!!!
It's not even so much about thinking someone is cute! I mean most of the time I don't really think spiders are cute but I still don't want to hurt them and I'm trying to help them. I think the worst thing is when people think (that's what I've heard) bunnies are cute and they play with them and then after some time they eat them!!!!!!

that's like being some psycho killer!!!! that is totaly sick!!!
It seems not all domestic cats kill. This link is interesting, if somewhat macabre:
https://theoatmeal.com/comics/cats_actually_kill
My father used to have many cats and none of them killed anything. Of course being so fat and well-fed that they were incapable of rapid movement might have had something to do with it.
Well I know my cat doesn't kill any animals!! Cause she never goes out alone!

well..... I think she may have killed some bugs...

but now I think she kinda understands that I don't like it... so she maybe pokes a bug a little or something but doesn't really kill it so I can take the bug and take it outside

one time this summer she was poking something on the floor on the balcony and I looked and I saw a spider lying there and first it was still moving but it was on it's back and then I picked it up (with a paper) and put it up in some plants and it wasn't moving at all and then I thought it was dead, but when I came back after just a few minutes it was gone (and no, my cat can't get up there and it was dark already so no birds or anything could get it) so it was still alive, I was really happy, and it happend the same with some other bugs that at first I thought were dead but then they started moving and flying again. (And I always put them in the same place, maybe that place has some magical healing powers!!

).