How does the "reverse syntax" in Middle English work?

First, a point of clarification. The "Therefore did Tristan claim justice..." passage that you quote seems to be from a translation made sometime around 1900 by Hilaire Belloc of an originally French text. That is well into the Modern English era; the period called "Middle English" is generally considered to have ended around 1500.

A Middle English text about Tristan looks like this:

I was at Erceldoune,
Wiþ Tomas spak y þare;
Þer herd y rede in roune
Who Tristrem gat & bare,

(Sir Tristrem, Auchinleck manuscript, National Library of Scotland)

That said, the grammar of your examples does have something in common with the grammar of Middle English. All of your examples use "V2 word order". This is a feature that still exists today in certain other Germanic languages, such as German and Dutch. With this word order, the finite verb (the simple verb form that is inflected for present or past tense) is placed right after the first element of the sentence (or clause). (Not after the first word; a phrase can also occupy the first spot in the sentence. Also, note that for the purposes of this rule, the coordinating conjunction "and" doesn't count as an element of the sentence/clause.)

"Therefore did Tristan claim justice..."

"therefore was he careful to fail in nothing..."

"Quickly did he go...

"Smart is he who..."

And in the Middle English sample that I quoted:

I was at Erceldoune,
Wiþ Tomas spak y þare = [With Thomas spoke I there]

The prepositional phrase "with Tomas" occupies the first slot of the second clause in the preceding sentence.

You can find more information about the use of this word order in different stages of English if you look at the linked Wikipedia article or Google the term. V2 word order had started falling out of use already by the end of the Middle English period. In the modern English era, this word order becomes so much less common that it starts making more sense to describe sentences in terms of "inversion" triggered by special elements or special meanings (as mentioned in geekahedron's answer), rather than in terms of the position of the verb relative to the first element (whatever that might be).

It is harder for me to describe how the meaning is changed by putting one part of the sentence first vs. another part. I'm sure this has been studied, but I don't know what the main patterns are.

These are examples of subject-auxiliary inversion, where the subject switches places with an auxiliary verb (which, here, includes forms of "to be").

Generally, you see inversions in questions, conditionals, negative statements, and elliptical sentences using "so," "such," "as," or "than." The examples you have cited do not fit any of those specific use cases, however, and such optional inversions are grammatical but only seen in archaic texts.

The meaning of the statements is not changed by the inversion, but the emphasis may be different, or used for anaphoric effect (as in the Beatitudes, for example).

If the verb is not an auxiliary verb, it is called subject-verb inversion, and the same rules of inversion (and optional inversion in older English) apply.

As someone (in a comment that has since been deleted) pointed out, this is a vestige of V2 word order, seen in other Germanic languages and common in Old English until a transition to SVO in Late-Middle English.