The question is on phrases like the ones that begin the following sentences.
(a) Her chin on the tabletop, she looked at me.
(b) His back turned to the door, he was writing a letter.
(c) His time being short, he turned to what he thought was the most critical question.
(1) Is there a single grammatical term for such phrases? We may say that the phrases in question
consist of a noun and some verbiage following it,
do not connect to the rest of the sentence by means of a "grammatically revealing" connector (for example, a preposition governing the noun) and
can easily become a sentence in which the noun is the subject and the other verbiage the predicate. ("Her chin was on the table top." "His back was turned to the door." "His time was short.")
(2) Are they noun phrases? (Some of them may be another kind of phrase as well, but is it true that they are all noun phrases?)
(3) Which case are the nouns in (e.g. nominative, accusative)?
On (3), I suspect maybe not the nominative (not always anyway) because, when there is a pronoun involved, we want to say something else:
With him gone, there was no chance now.
Also in German we seem to get cases other than the nominative, as in the following.
Am Fenster saß an einem Schreibtisch, den Rücken der Türe zugewendet, ein kleinerer Herr, der mit großen Folianten hantierte, die auf einem starken Bücherbrett in Kopfhöhe vor ihm aneinandergereiht waren.
I am told that the phrase is considered a participial construction in German (an answer I got from German StackExchange). (I guess it's a separate question whether a "participial construction" is necessarily not a noun phrase.)