In traditional grammar the role as plays here is called a subordinating conjunction. In more recent grammars it is called a subordinator or adverbial subordinator.
It marks the following clause as ‘subordinate’ to the main clause, acting as an adverbial which modifies the main clause. In this particular case the subordinate clause is an adverbial of degree; the Russian law has disturbed the world of classical music to a greater degree than all but a few other recent political issues.
[III] - I ate the pie, and since then I have had a stomach ache.
I think this is the most natural comma placement: it divides the two independent clauses, which is what commas do.
I don't know who wrote your Sentence I (SI). To me, putting the two commas here is unnatural. It represents to me an unnatural way of expression. It also means there are two pauses within five words and two pauses in a straightforward sentence of 13 words. I don't know why anyone would want to slow down that much.
The two commas in SII make sense, because they separate a unitary phrase (since then) from the rest. But although this two-comma version works much better than SI, it's not as eloquently simple and natural as SIII. This is because SII also introduces an unneeded double pause in a thirteen-word sentence that needs, at most, one pause. (Note that my use of commas to separate at most is similar to II's comma use.)
You could also dispense with the 'since then' and write the remaining with either no comma or one comma:
[SIV] I ate the pie and I have had a stomach ache.
To me this expresses both thoughts as one unit containing two facts. It does not really stress a causal or resultative relationship between the two actions/facts. Also a comma is just plain unnecessary because now the sentence is unencumbered by the 'since then', and it has only eleven words. I mean it is short and consists of two short independent clauses.
[SV] I ate the pie, and I have a stomach ache.
This goes back to the simple connection of the two independent clauses with a comma immediately after the conjunction, as in SIII. This is very frequently done, and at least helps the reader parse the sentence if not also slow down a tiny bit. (See how I used only one comma in that last sentence, and only to separate the two clauses? Oops, I just did it again.) SV probably does not present the two actions in terms of one unit of dual-facts like SIV. Perhaps because the pause also allows a millisecond for the reader to make some causal or resultative assumption between the two clauses--even though the comma by itself does not do that.
Best Answer
Here are your examples with "before":
Preposition. Reason: it takes a noun, breakfast, as its object. "Before breakfast" is a prepositional phrase, which functions as an adverb, modifying "Mac had a headache."
Preposition. Reason: it introduces a prepositional phrase, ending in a noun, as in the previous sentence. "Eating" here is a gerund, serving as a noun. Notice that you could say "Eating breakfast starts your day off right." The verb in that sentence is starts, which is singular to agree with eating, the subject.
Conjunction. Reason: it introduces a whole clause that has its own subject. "He ate breakfast" could be a whole sentence. "Before" converts it into a subordinate clause.