[Casiopea]

"be asleep" is a set pair. The first word "be" is a linking verb, and the second word "asleep" is an adjective. As for "go asleep/went asleep", it sounds odd to me, sorry. I'd use, "She went to sleep fast."

"fast" can be an adjective or an adverb. When "fast" modifies a noun, it functions as an adjective,

Adjective: She has a fast car. The car is fast. ('car', 'The car' are nouns)

When "fast" modifies a verb or an adjective, it functions as an adverb,

Adverb: She is fast asleep. She fell asleep fast. She went to sleep fast.

When in doubt, replace "fast" with an adverb, and if the sentence sounds OK, then "fast" is an adverb, like this,

She fell asleep fast/quickly. ("fast" is an adverb)
She is fast/completely asleep. ("fast" is an adverb)
She has a fast/quickly car. (ungrammatical) "fast" is not an adverb here.
She went to sleep fast/quickly. ("fast" is an adverb)