At the end of the movie there was something under Joshua's bed. It looked like a glove to me. What was it? If it was a glove, it could have possibly been the uncle's (given it looks like a piano glove) indicating a sign of sexual abuse.
I thought that was Lily's binky. I could be mistaken, but it would make sense for it to be under his bed/in his possession. Even though the father did not realize it was missing until they were at the park, he suspected Joshua took it-if the binky is what is under the bed, then the father was correct.
It was a rubber glove. It shows the hamster which has been taxidermied, indicating he was clearly killing the animals and he took his little fascination with Egyptian mummification...far.
His parents caused him to resent his baby sister for a while because they acted like she was the only child of theirs they cared about. They seemed to forget they even had a son, especially the mother. So if he did want to get rid of his sister, I can't say I blame him because that's the only way they would give him love and attention again. Although I honestly don't believe he truly hated her as much as he was jealous of her. I don't think he truly wanted to kill her in the end; he just wanted to get himself and his sister away from his parents and live with his uncle, who was the only person he felt understood him and loved him.
He did sing to his uncle, you & me, the two of them, and didn't include Lily in the song? He actually showed some affection to her and did include her in his escape plan. I do think it was his plan to take the binky to upset his father through Lily.