It's not the corgi we meet first; it's an attacking Dobermann who's on top of a man lying down, biting him or trying to. ... We don't actually see the dog die, we only hear a shot and see the man who shot (at) it soon after, looking more or less fine, so it's to be assumed that the dog was at the very least badly enough injured to be unable to continue attacking.
Twice a person comes on to someone very strongly, physically pressing up/rubbing against the person. In the first case, the "approach" is very unwelcome (but there's only slight discomfort for comical effect); in the second not entirely so.
Someone has one half of his moustache cut off with a rapier, so precisely that the moustache flies through the air in one piece, and his lip is smooth and unharmed. :D Apart from that, shaving is a recurring subject throughout the movie, but no one ever actually gets shaved, only lathered. I'm not sure if anyone gets cut.
Not visibly or audible, but someone says he just cracked a rib. Also, a person is thrown out a window and lands heavily with someone else on top of him, and it's rather implausible that he doesn't break some bone or other, but we don't know.
A woman travelling with a baby and a man who appear to be her child and its father has sex with a stranger in the toilet on a plane. Other than that: Not that we know of for sure (someone says something that implies that they've cheated more than once, but it's probably just teasing). However, two people act as if they might very well cheat with a specific person in the very near future. Heavy flirting and innuendo.
I don't think anybody dies by fire, but several people's clothes catch on fire in the first scene. We see them running around in a panic, their backs and arms on fire. In another scene, a rapidly spreading fire results in a chemical explosion, but I'm almost 100 per cent sure everybody in the room makes it out in time.
No, but there are two scenes in which someone comes very close to having their finger chopped/cut off, and someone is shot in the hand (palm) and has someone pushing their finger into the wound.