In your particular example, both "is sleeping" and "is asleep" are fine. You can use either. However, "sleeping" can be either a gerund or present participle, which means you can use it as a noun, adjective or in a verb form (continuous). "Asleep" is an adjective or adverb. Morgan has already given some examples where you use one and not the other. In fact, you would say "a sleeping child" and not "an asleep child".

Yes, you may use either one. They are interchangeable.

Generally, "asleep" always follows a verb. The child was sleeping/asleep when I walked into the room. Their child is a nuisance, so I am glad he is asleep/sleeping at the moment! Sleeping well is good for you. (not: Asleep is good for you) Although sleeping in the afternnon is comfortable, it isn't always practical. (not: Asleep in the...)