Stay awake, is awake here an adjective? my dictionary says so! but it sounds like an adverb.
"Awake, alive, asleep etc" are predicative adjectives, ie they can only be used after linking/copula verbs like to be and similar verbs. They can't be used before a noun.
The prefix a- was originally a preposition like in, on or of. Asleep means in (the state of) sleep.
By the way there are several prefixes a- with different meanings eg anormal meaning not normal. http://www.yourdictionary.com/a#websters
The underlying meaning of stay and remain is "continue to be" and so they function like copular verbs, which connect an adjective (or other noun complement) to a subject.