In each of these cases, 'sleeping' is a present participle. A present participle can function as an adjective, as in the first sentence - 'the sleeping child.' It can also combine with the verb 'to be' (am. is, was etc) to form a continuous tense, as in the third sentence, 'The child (who) is sleeping'. The second sentence is a reduced form of the third sentence: 'The child sleeping' is a reduced form of 'The child [ who is] sleeping.' NB Be careful not to confuse the present participle with the gerund. Gerunds and present participles have the same form but different functions. A gerund is a noun form of the verb - for example 'He likes sleeping' or 'Sleeping too much is bad for you'. These are gerunds, not present participles. Many present participles can be used as adjectives, but not all. I hope that helps.

I'm not sure about your first question. You seem to have answered it yourself. For the second question, a "[verb]ing" form can be either a gerund or a present participle. These work in completely different ways, so it's impossible to confuse them. Don't be tricked because they look the same. As a participle, [verb]ing can work as an adjective, as part of a continuous form, or as the beginning of a participle phrase. In all cases, the present participle cannot work alone, and needs another word in the sentence to function.