The dog, Seamus, who is one of the main characters, is NOT killed but since the book is set over the course of ninety years obviously the dog cannot live that long and the very end we are told about his passing in a lyrical, moving, but non-traumatic passage. The book has been praised for the way it has handled this and it definitely a book for dog-lovers.
This book is set in the near future and one scene is set at a concentration camp full of children who are being taunted by soldiers. Several children are killed in this scene but there is no graphic detail about it; it is handled as if it happening offscreen.
There is not a hanging but a man is caught in a trap that hangs him upside down by his leg. He dies from these injuries but this happens before his body is found by the main characters.
No, but the book is set in a near future where it's imagined that LGBTQ relationships have been outlawed. The main character is gay and on the run from forces that would imprison or kill him for being gay.