Do horses deliberately kill other horses?

Very rarely. Most incidents of a horse injuring another horse fatally are accidents - they're large, powerful animals. Horses generally do not fight to the death, but they can fight quite badly. I know a mare who got into a fight with more than one pasturemate, which culminated in her killing another horse and being condemned to being turned out alone for the rest of her life (clearly she should have been isolated sooner, but she had previously been okay with a different companion).

Horses can be aggressive, but generally not murderous. Most horse fights happen when animals that don't get on are put together in a smaller pasture where they don't have space to leave each other alone.