"You talked a lot."
Is it absolutely incorrect to swap the order, as "Beszéltél sokat? From what I'm coming to understand, this would put the emphasis on the verb, perhaps in a situation where someone says "I ate a lot" and someone else says "you talked a lot."
Not really. Beszélget (in this form it should be beszélgettél - you chatted) is more of a forth-and-back thing. But in this sentence it's pretty clear that only one person does the talking.
Grammatically speaking, the "a lot" part is viewed as the object of the verb "talked". Objects in Hungarian take the accusative case, which is marked with the -t, and a linking vowel where needed. So sok ("a lot" as the subject of a verb) becomes sokat ("a lot", as the object of a verb).
Sok isn't an adverb here, it's a noun (or at least something very similar). "A lot" is what's being talked.
Adverbs, on the other hand, do generally not change in Hungarian. Since they directly modify other words, they cannot stand by themselves and thus get no case endings. "Nagyon köszönöm."