"The irate customer asked for the chef."
The irate customer asked something. (Noun phrase?)
Since you can fill in something in place of 'for the chef,' does that mean it is a direct object and an adverb at the same time?
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Someone is indeed a noun phrase acting as direct object, but it in no sense stands in for for the chef.
Your first example uses the phrasal verb ask for, which means to require or try to get something (or somebody).
Your second example uses the different verb ask, which means to make a request or pose a question.
Look at the difference between
What did the customer ask?
What did the customer ask for?
The first one is answered by the expression of a question (eg where the chef was), the second is answered by the thing (or in this case, person) that the customer was seeking (eg the chef).
There is a third possibility if the direct object is a person, or "who":
Who did the customer ask?
Who did the customer put a question to (or make a request of)
and is again quite different from
Who did the customer ask for.
Ditto Colin on the specifics here.
But more generally, it looks like you're thinking that because you can substitute X for Y in a sentence and it still makes sense, that this must mean that X and Y are the same part of speech or otherwise equivalent. This just isn't true. There are many different ways to construct a sentence. It's not like there's one simple pattern with only one possible type of component in each place.
Like, I can say, "Bob walked quickly" and I can say "Bob walked the dog." The fact that I can replace an adverb with "the dog" doesn't make "the dog" an adverb, etc.