1. The photo Arthur is holding has a note that says 'I love the way you smile. - T.W.' This implies that Thomas Wayne and Penny had more of a relationship and Arthur might not have been adopted.
After Arthur's mother dies, he finds a picture of her in her belongings with a note from Wayne. The film doesn't elaborate further, but it could mean that Wayne was lying when he said nothing happened between them. With the kind of money and power that Wayne commanded, he might have rigged things so it only appeared that Arthur was adopted.
2. Arthur was committed to hospital soon after he stopped taking his meds, and imagined his own rise and the fall of the Wayne family in his mind.
At the end of the movie, he's handcuffed and speaking to a psychiatrist. He can't stop laughing, and when she asks why, he says it's a joke she wouldn't get, all while the camera cuts to shots of Bruce standing over his dead parents. There's an ethereal, dreamlike quality to the whole sequence, leaving us unsure if any of the things even happened.
3. The time jump at the end is much longer, and takes place in the future where Batman already exists and got the Joker put in Arkham.
Since it's not explained how he was caught, it could be that Arthur was committed to the hospital many years later by Batman himself. It's also possible that in the process, Arthur discovered that Batman is Bruce Wayne and realizes he created Batman the day he became the Joker - which he would obviously find hilarious. That's why he tells the psychiatrist she won't get the joke.
For now, we don't really know if we'll be getting any answers, or even a sequel. All we can say is, good freakin' job Joaquin!