In the Series 2 episode 'Blood is Thicker', the Flint brothers kill their dog and stuff it as a gift to Dr Ellingham for helping with their father. In the Series 3 episode 'The Admirer', Dr Ellingham accidentally runs over a dog and wraps it in newspaper, although the dead dog is not explicitly shown.
Some spoilers are featured in this comment. Martin himself is seen throwing up multiple times on screen, I don't have the exact episodes or time stamps but I know that most of it happens in season one, due to a severe blood phobia. I can't remember it too well, but as far as I know, no actual vomit is seen in terms of Martin. Otherwise, there are various instances of characters throwing up, including one episode with a boy who receives an injury early on and is seen later in the episode laid in bed and covered in his own vomit.
In the Series 3 episode 'The Admirer', Dr Ellingham accidentally runs over a dog and wraps it in newspaper, although the dead dog is not explicitly shown.