When you're talking about gazing toward the sky, it's an adverb. You can tell because it doesn't need an object. You can just tell someone to "Look up." Also it can be followed by a preposition: "Look up at the sky." Whenever you have two words that can be prepositions that are together, the first one is functioning as an adverb. When you "look something up", it's a preposition that needs an object.

Depends on the sentence where you found it. "Look up" may mean "respect" so it won't indicate direction in such case.

Both. "Look up the chimney to see if there's something stuck there." (preposition) "Look up his name in the telephone directory." (adverb) "Look his name up in the telephone directory." (adverb) In these last two examples, "in the telephone directory" is also an adverb phrase.