Some of the many doors stood wide open with lads carrying sacks and buckets in and out, and some were half closed with horses’ heads looking interestedly over the tops.
I have a question about the function of 'with lads carrying sacks and buckets in and out'. Is it absolute nominative construction? If it is not then what is it? And is the phrase 'with horses’ heads looking interestedly over the tops' the same construction?
No: absolutes are always non-finite clauses, not preposition phrases. I'm inclined to analyse both with PPs as adjuncts in clause structure. Note that the prep with has as its most basic meaning the notion of accompaniment.
At the head of the expressions is the preposition "with", so they are preposition phrases. The preposition "with" has a non-finite clause as its complement. But that is not relevant to the analysis.
[Some of the many doors (subject)]
[stood (verb)]
[wide open(Adj complement)]
[with
--[lads carrying sacks and buckets in and out(non-finite clause)]
(prepositional phrase)],
