Jonathan Robie wrote:
Is there a simple subset of discourse analysis that is agreed-upon, and relatively straightforward?
Discourse analysis (= text linguistics) is still a relatively new discipline and it deals with topics that are open to subjective analyses, so I don't think there will ever be consensus.
But let me try to give a simple introduction. The smallest meaningful bits in the grammar are called morphemes. They are put together to produce a word. There is usually a root and one or more affixes. An affix can be a prefix or a suffix or even an infix. Words are put together to make phrases, phrases make clauses, clauses make sentences. Traditional grammar basically deals with everything up to the sentence level. Discourse analysis deals with higher levels, that is, how sentences are put together to make paragraphs which again make sections or episodes and we end up with a whole discourse/text. Any language has a number of particles or affixes that cannot be adequately explained unless you go beyond the sentence level. For instance, καί can function at the word level to coordinate words. It can also function at the phrase and clause level to coordinate two or more parts. When it coordinates sentences or paragraphs it becomes more complex and is a topic for discourse analysis. A particle like γάρ may be given the meaning "because" or "for" and treated as a conjunction between clauses, if one is limited to traditional grammar. But if we study how the word is used to connect sentences at the higher level, we get a much more satisfactory explanation. This particle supplies further background for one or more words in the previous context, almost always the previous sentence. Similarly δέ can not be explained within a system that does not go higher than the sentence. It cannot be explained by an English translation. One has to look at the function. Basically, it introduces a change of some kind, but the exact kind of change depends on the word that precedes it and the context.
The tracking of participants through a story is another common topic of study in discourse analysis. That is needed to properly account for the use of the definite article and demonstratives. Another topic is the developping of a story through setting, building-up units, peak and release/denouement or closing. This overlaps with literary studies. This is needed to properly study aspect and some other areas.
The function of word order is not really a matter of discourse, since it does not go beyond the sentence. It belongs to pragmatics. However, when we study texts we are better off if we can use insights from both traditional grammar, semantics, pragmatics and discourse analysis.
Different schools use different technical vocabulary. A few decades ago it became popular to talk about topic and comment, dividing every sentence into two bits. I have never personally felt that this was useful, so I don't use it, and I think I get by quite fine without it. The word focus is also used to mean several different things, so it is not particularly useful. I prefer to talk about relative prominence which is not dichotomistic. My philosophy as a practical linguist and translator is that the simpler the rule, the better.
Those are my 2 cents.