I remember the occurrence of this type of construction from many moons ago, but not its particular reference :
Absent the doctor, the paramedic pronounced the victim deceased.
I’m not clear as to what is going on in the phrase ‘absent the doctor’.
There could be an ellipsis here :
Absent [being] the doctor …
The verb ‘absent’ can be used :
transitively : Her mother absented her from school.
or intransitively : The workers absented from work.
But I am not clear whether ‘absent’ is a verb here.
The same kind of thing happens in :
The patient dead, there was no more to be done but sign the certificate.
I am also interested in what role the definite article is playing.
For one could also say :
Dead the patient, the room became hushed.
Is there such a thing as a ‘hanging’ adjective ?
You’re looking at the wrong definition for absent. The OED page you want is the preposition one (requires subscription), which defines it as:
orig. and chiefly U.S. Law.
In the absence of, without.
So it’s not an adjective, although this use of the word was derived from the adjective form. The definite article is only there because it is needed for "doctor"; other sentences wouldn’t use an article: "Absent federal legislation upon the subject…" (source).
However, I will note that there is a (completely unrelated) construct where the adjective always goes before the (indefinite) article:
It’s too big a mess.
This is known as the Big Mess Construction.