It's heavily implied Lord Farquaad killed Mama Bear and had her made into a rug. Early in the film, we see Mama, Papa, and Baby Bear all together as they're forced to relocate. However, later on, only Papa and Baby are seen. During a scene in Farquaad's bedroom, we see a bear rug resembling her. A frog and a snake are also humorously made into balloons, and we see the characters eat (cooked) dead rats. A fish and a bird are also killed for consumption.
In the dance party ending, Shrek tries to crowdsurf, but ends up landing on a small dog instead. While the dog getting crushed isn't shown, we do hear the crowd gasping and see Fiona's reaction. Largely played for laughs. However, in another shot shown just before the credits, the dog is seen alive and well.
In one scene, a police dog survives an explosion, apparently unharmed. In another scene, it serms like something bad might happen to a dog barking in a car, but it does not come to any harm.
A dog dies after falling and being impaled on a clump of knitting needles.
There are two pet dogs in this film. One dog is presumed dead after it goes silent after being left outside. The other dog is stabbed to death with a barbecue fork after trying to attack two children.
A pet wolf is knocked out by one of the main characters who is a big muscular man that could have easily killed him with his bare hands if he wanted to. The man however says that he doesn't want to kill the wolf, and only defended himself. The wolf is last seen unconscious but alive.
A dog appears and is unharmed. (A box is dropped on a scorpion, killing it.)
Pet dogs are seen briefly, barking at bad guys. They are not harmed.
Snakes on a Plane
A dog is eaten whole by a snake and it is implied that a cat is eaten also.
A dog swallows something it shouldn't and is shot at but is not harmed.
Some dogs fall through the ice but are rescued.
A team of sled dogs end up in some perilous situations but they survive.
So Dear to My Heart
Danny the lamb is lost for a short time but is recovered unharmed and does not die.