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.
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.