A call was sounding.
This is not an existential construction. There is no dummy subject. We can consider this clause to consist of an ordinary subject (a call) and an ordinary intransitive verb (was sounding). The verb employs the active voice, past tense, continuous aspect and indicative mode.
At a glance, it might seem oddly redundant: a noise was making noise. However, a call isn't simply a noise. It might not be (and, in this context, is not) a literal noise at all. A call is a request for a response; it requires an answer. A request for his presence was making itself known.
It is a complete independent clause, and it could stand on its own as a sentence.
A call was sounding in the forest.
The prepositional phrase "in the forest" modifies the verb "was sounding".
A call was sounding deep in the forest.
As I parse this, the word "deep" modifies the phrase "in the forest". I consider this example of "deep" to be an adverb.
Others might label "deep" differently, as an intransitive preposition or a particle. Another possible parsing is that "in the forest" modifies "deep".
In either case, the phrase "deep in the forest" is grammatically coherent. I'll let others argue about which word is the head of the phrase or how its dependency structure should be understood. One way or another, the entire phrase "deep in the forest" functions as an adverbial prepositional phrase. As a unit, it modifies the verb "was sounding", just as the simpler "in the forest" does in my earlier example.
Deep in the forest a call was sounding.
Here, the adverbial prepositional phrase is placed at the beginning of the clause. That changes the rhythm of the clause and gives more emphasis to the leading word "deep", but it doesn't change the grammatical structure.
This entire independent clause is, in the original text, just one part of a long, convoluted compound-complex sentence. It could have been a sentence on its own, but in the original text it is not on its own and it is only one part of a sentence.