there is a cerberus boss that is fought and killed, however, this isn't exactly an emotional moment and the cerberus is barely canine beyond its appearance.
There are many attacks which can be used to light enemies on fire, and your protagonist can also be hit and burnt with firey attacks. However, it's not especially graphic, just simply another way you can hurt things/be hurt
The character Griffon, who is a bird-like demon with a human voice and intelligence, is hurt in comical ways in some cutscenes, due to his annoying nature.
the demon goliath uses an attack where it fires blasts of energy from its second mouth after eating debris. It's not exactly vomit but it may be of note.
whilst it is technically not suicide as this action does not kill him, Vergil is shown impaling himself with his sword as a last resort. Dante also does this later on.
there is a LOT of blood, the whole plot basically revolves around it. even the environments of the game are made up of blood and gore due to the demonic tree that takes up most of the game's world