An article discussing -ing forms and post modification.
About these pesky –ing forms. In your example about the robber and the note, the phrase demanding $1000 – technically a non-finite clause formed from an –ing participle – provides us with more information about the note. That is, it post-modifies the noun. The complete noun phrase reads a note demanding $1000, with note being the head of the noun phrase, the article a being its pre-modification and the –ing clause its post-modification. Together the noun phrase functions as the object of the verb gave. The entire sentence instantiates a Subject-Verb-Object (SVO) pattern.
Other common ways we post-modify nouns include preposition phrases (the man on top of the ladder), and relative clauses (the woman who you gave your phone number to). As you correctly observe, the non-finite clause a note demanding $1000 could be reformulated as a relative clause a note which demanded $1000, but, strictly speaking, it is not reduced (or elliptical) because a note which was demanding $1000 would sound ungrammatical. Another example: I bought a picture resembling the one you have can’t be expanded to I bought a picture which is resembling the one you have. But the man sitting in the corner can, of course, be expanded to the man who is [or was] sitting in the corner. The point is that not all non-finite clauses are simply 'reduced' finite ones.
Incidentally, pre-and post-modification of nouns is one of the most problematic areas of grammar for learners, yet you seldom see it dealt with full-on in teaching materials, apart from the (grossly exaggerated) attention given to relative clauses. The use of –ing and –ed forms to post-modify nouns is much more common than using relative clauses: The train approaching platform 6 is the 10.36 for Beaconsfield…, Baggage found unattended will be removed and may be destroyed etc. The most common ways of post-modifying nouns are preposition phrases, and of-phrases: the shop on the corner; the land of our fathers, etc.
As for your second sentence She walked out of the room smoking: this is clearly not an example of noun post-modification, because it is not the room that was smoking, but the woman. That is to say, smoking doesn’t form a syntactic unit with room. It is a kind of add-on to the whole clause she walked out of the room. It provides extra circumstantial information that could, of course, be reformulated as a finite clause: She walked out of the room and she was smoking.
You are almost right in saying that smoking is an adverb. In fact, it’s another non-finite clause formed from an –ing participle. But it functions as an adverbial. An adverbial, note, not an adverb. So, the syntactic pattern of your second sentence is SVAA, where two adverbials follow the verb (out of the room + smoking).
What’s the difference between an adverb and an adverbial? (I hear you ask). 'Adverb' simply describes the word class of words, like happily or well or not. But it doesn’t tell us how the word functions in a clause, in relation to other elements such as subjects and objects. 'Adverbial' is the term used to describe the way some words, or groups of words, function, irrespective of their word class, when they add circumstantial information to the clause. Thus, all the following underlined words, phrases, or clauses are adverbials, but each has a different form (which I’ve put in brackets):
She played happily. (adverb)
She played quite well. (adverb phrase)
She played in the park. (preposition phrase)
She played all morning. (noun phrase)
She played, laughing all the time. (non-finite –ing participle clause)
She played, while I watched. (finite clause)
Again, the range of possibilities in selecting an appropriate adverbial in order to extend the meaning of a sentence is seldom if ever dealt with systematically in coursebooks – to my knowledge. Unsurprisingly, though, the section on adverbials is by far the largest in the Longman Grammar of Spoken and Written English.