Can an adverb be a noun at the same time?

A noun can be used as an adverb as well in cases that do not involve the locative.

 "He stood at the door **hat in hand**." 

Clearly, hat modifies stood by telling the manner in which he stood.
Grammatically this is similar to the adverbial function found in "He ran fast."

The OED lists hat as a verb and a noun. It does list hat in hand under phrases and locutions, but does not analyze the grammatical category of the word, hat, in that case.

Of course, one could imagine many more examples: "He stood at the door chair on shoulder,*face aglow*,pockets empty, etc. There is really no limit to such usage of nouns as adverbs.
In the right context, English can handle it very well.