I know asleep and sleeping are interchangeable in many cases. But in these situations, I am not sure.
I read stories to my son after he went to bed. After reading the stories, I stayed for some time and tucked the quilt for him. When I came out of his room and met my wife, she asked "Is he asleep?" or "Has he fallen asleep?", or "Is he sleeping?"
In the morning, I woke my son up before he went to school. But later I found he did not get up. His eyes were closed. I asked, "Are you still asleep?" or "Are you still sleeping?"
Which is better in each situation?