First, some corrections: one cannot normally say the following, which is not an acceptable constructions in contemporary English:
*Dead the patient, the room became hushed.
Also, the intransitive use of the verb absent is nowadays regional (according to OED, 'chiefly S. Asian'), and the transitive, non-reflexive use is rare. In contrast, the phrase absent [something] is not rare. This is an indication that absent is unlikely to be a verb here.
As Robusto says in the comments, absent is a preposition in your examples.
It is true that, grammatically, the following can be a complete clause:
Absent the doctor.
Here we take absent as a transitive verb (which is a rare usage nowadays, as I said above). Since the subject is lacking, this would have to be in the imperative. So
 Absent the doctor, the paramedic pronounced the victim deceased.
would have to be interpreted as something like
Because the paramedic pronounced the victim deceased, you should keep the doctor away.
This is a possible reading, but a highly unlikely one.
Your other suggestion was that there is an ellipsis, so that the full sentence is
Absent being the doctor, the paramedic pronounced the victim deceased.
Strictly grammatically, this is OK, being comparable to
Absent being able to smash through the glass, an intruder will have to somehow force the window open. (source)
But the meaning would be a bit strange, since it would be something like 'because the paramedic wasn't the doctor, he pronounced the victim deceased.' I suppose this is not strictly impossible, but what we normally mean by  is rather something like
Because the doctor wasn't there, it was the paramedic who pronounced the victim deceased.
And in that reading of , absent is a preposition.
Here's CGEL (p. 610):
 absent adjacent consequent contrary effective
exclusive irrespective opposite preliminary preparatory
previous prior pursuant regardless subsequent
These items all qualify as prepositions by virtue of being able to occur as head of an adjunct with no predicand, as in such examples as:
 i [Absent such a direct threat,] Mr Carter professes to feel no pressure.
ii [Right adjacent to the church] there is a liquor store.
iii [Consequent on this discovery] there will doubtless be some disciplinary action.
iv He had not been seen in the area [prior to this].
v The plan will go ahead [regardless of any objections we might make].
Irrespective and regardless, although historically adjectives (as reflected in their morphological form), are now virtually restricted to the preposition category. The other items in  occur in addition as adjectives in attributive function - compare absent friends, the adjacent building, the consequent loss of income, etc. In the case of absent, effective, exclusive, opposite, preliminary, and preparatory, there are also complement uses which are clearly adjectival: Five of them were absent; The film was very effective; This club seems very exclusive; and so on. With the others, however, there is little reason to distinguish the complement use from the clearly prepositional adjunct: compare, for example, This was prior to the election and This happened prior to the election. NP complements are licensed only by a few of the prepositions: opposite (opposite the church) and, in specialised registers, absent (as in [20i], "in the absence of") and effective (Effective 1 July the fee will be increased to $20).
The patient dead
We are considering the construction
 The patient dead, the room became hushed.
There are three main possibilities for the analysis of the patient dead: that it is an adjective phrase (AdjP), a noun phrase (NP), or else that it is a non-finite clause with ellipsis.
There are two possibilities how it could be an AdjP. One is where dead is the head and the NP the patient a pre-head modifier. However, according to CGEL, 'adjectives take only a highly restricted type of NP [noun phrase] as pre-head dependent' (p. 1656). On pp. 549–550, CGEL lists the allowable types of NP pre-head dependents: they are either measure phrases (as in three years old, five centimetres thick, a foot wide, two hours long, …), or else quantificational NPs (as in a bit lax, a smidgin overripe, a tad greasy, a trifle shy, plenty big enough, …). Some quantificational ones occur only with comparatives (a great deal smaller, a (whole) lot different, lots better, heaps worse, …). But in all cases, notice they are truly complete noun phrases (NPs), meaning that they include a determiner when one is necessary.
Another way how it could be an AdjP is that it is an adjective-centred compound adjective. Some examples of those are accident-prone, burglar-proof, oil-rich, snow-blind, tax-free, etc. But this interpretation just doesn't fit. Clearly we are not talking about a particular kind of dead, the 'patient' kind of dead; clearly, we are describing a situation resulting from a patient having just died.
As far as it being an NP, this would mean that dead is an adjective used postpositively. Also, this NP would function as a supplement in that sentence. However, this is not how supplement NPs enter sentences. NPs as supplements enter either appositively, e.g. The murderer, the man with the scar, will be arrested soon; A Sevfert galaxy - a galaxy with a brilliant nucleus - usually has a massive red-shift (CGEL, p. 1357) or else ascriptively (e.g. Robert, no genius, is applying for a scholarship to Harvard).
Having exhausted options that don't postulate an ellipsis, at last we may consider the possibility of an ellipsis. Most likely it is exactly what the OP suspected:
 The patient (being) dead, the room became hushed.
This is somewhat atypical, though, because (ComGEL, p. 910)
typically nonfinite and verbless clauses lack both subject and operator, and that their relation to their main clause can be explained if we postulate, in many cases, an ellipsis of these elements, the identity of the subject being recoverable from the main clause.
Nevertheless, this seems to be our best option.