Yes, the neglect of the rich white christian (presumably) american cop loving family has led to the wistfulness of their child to care for his dog who he then abuses in neglect out in the woods.
No, there is no abandonment. The mother had a rough childhood with strict, unfeeling parents shown in flashbacks about her childhood dog, but there's no abandonment, abuse or neglect.
A woman believes that a man is abusing animals mentioning how a lot of dogs come in but none ever comes out, but it turns out that she's not all there and this man is actually rescuing dogs.
Gonker does not die, but there are brief mentions of a few other people having lost their dogs as well a scene where parents tell a girl that her dog got away and was hit by a car and died as a result.
Fielding is in the hospital at the end of the movie but it’s nothing sad or triggering SPOILERS: his parents bring Gonker to him in the hospital and it’s a happy scene
Drunk guys at a bar make fun of Fielding being sad about his missing dog and are looking for ac fight, but his dad intimidates them into leaving without violence.