SO at the beginning of a sentence is a summing up device(*So began the fight.) but equally an inducing agent preparing us for the narrative that follows(*So goes the rumour. There was a ...)meaning 'like this'.
As a parts of speech, SO is an adverb or coordinating conjunction.
* I am so sorry.
* He was ill, so he could not attend.
It means /like this /thus /in the manner /to the extent(relevant meanings only)
An adverb may be placed at the beginning of a sentence for modifying a whole sentence (anaphoric pro-form)or simply for emphasising. Take for example:
* Down went the Titanic.
In the example of preacher, "so" means to me "like this" and that of fight,'thus'. SO is ADVERB in both the examples, nothing more.
Subject-verb inversion is not that common as is subject-auxiliary inversion. Still it is has evolved into a distinct pattern in which the verb should not necessarily be an auxiliary verb. Inversion is not limited to auxiliaries only. Cf.
*Beside the bed stood a lamp.
Admittedly rupture of the canonical order takes place mostly in holy books, Ecclesiastical writings, short stories or literal pieces.
It is said that introductory adverbs or adverbials are used with 'be'/linking verbs (*So be it) and verb of direction ((*So can Tercy). 'So' in the sense 'thus'/'like this'can be used with almost all the verbs in inversion. Of course, necessity primarily dictates discretion.