Hello Hela
As for the last question, I prefer to take the whole of "look for" as a verb for the convenience of its use. On the other hand I take "pull down" as <a verb + an adverb> because we can say "we pulled it down" when the object is a pronoun. Anyway the opinions about how to parse those multi-word verbal phrases seem to vary among grammarians. To me, grammar is just a tool in learning languages and my choice comes not from which way is correct but from which way is convenient for me to learn and use phrases.
paco