Difference between "asleep" and "sleeping"

Solution 1:

Asleep is an adjective in Are you asleep? while sleeping is a verb in Are you sleeping?

Use asleep if you want to emphasize the state and sleeping if you want to emphasize the action.

Solution 2:

A predicate adjective is one type of subject complement.

A few adjectives can be only predicate adjectives, never attributive adjectives (i.e. he is asleep).

Note that the 'is' that appears in the questions "Is he asleep?" and "Is he sleeping?" is very different in grammatical terms.

This difference, and the fact that asleep is predicate adjectives, help you to understand which is better in each situation you have described.