How to identify licit sequences of doubled prepositions

Is a phrase like "go out from the kitchen", which has two prepositions in a row, appears to be good English, and might be paraphrased as "go out, and go from the kitchen." but I have doubts about something like "go out in the kitchen."

Can "go out in the kitchen" be paraphrased as "go out and go into kitchen"? If this is so, could a meaning of "go out and go home" be shortened to "go out home"?

Your premise is faulty.

In the first place, “Go out from [place]” all-by-itself is not idiomatic in Standard English (though I would not be surprised to hear that it is used in dialect). If you Google Books “go out from” or “went out from” you will find that almost every hit is either an allusion to a passage in the King James Version of the Bible or an old translation of some other work executed under the influence of the KJV. I imagine that the KJV translators, and their great predecessor Tyndale, were striving to represent a Greek or Hebrew idiom as literally as possible.

In the second place, the ordinary idiom is go out of [place] (“She went out of the kitchen”), in which out of is a compound preposition, like “up in heaven” or “over by the barn” — think of the CCR song:

Down on the corner, out in the street ...

There’s no call to parse this as an ellipsis involving two distinct phrases or clauses; it simply means “She left the kitchen”.

“Go out in the kitchen” is likewise a single, non-elliptical clause. “Jane, could you go out in the kitchen and fetch me my shears?”

“Go out and go home” is different. Here you have two clauses representing two consecutive motions, out of one place and thence to another place. But you cannot apply ellipsis because go out and go home are not parallel constructions: “out” is [an adverb acting as] a component of a phrasal verb, “home” is [a noun acting as] an adverb.