No, it isn’t being ungrammatical, deliberately or otherwise.

• It uses some old-style or poetic ordering at times for the sake of the meter (metri causa) or of the rhyme (rimae causa), but neither OSV nor CSV is especially exotic, let alone SOV or SCV.
• It is effecting a style from Early Modern English or even Middle English that was no longer current as of the mid 1800s when it was written, but was still remembered.

---

Historical Background

Christmas saw hard times under Oliver Cromwell’s Interregnum, when churches weren’t even permitted to be open on Christmas Day unless it happened to fall on a Sunday. Remember that Cromwell was a Puritan, and Puritans didn’t like the secular observations of Christmas, particularly their pre-Christian elements and their mercantile connections.

So before all memory of Christmases past was forever lost, during the later 1700s through the early 1800s hurried folklorists researched, assembled, and published as many existing Christmas carols as they could find still passed down by living memory in the oral tradition of the English countryside.

In 1822, Davies Gilbert published A Collection of Ancient Christmas Carols, with the tunes to which they were formerly sung in the West of England which contained some of these old forms. In his preface, he wrote:

The Editor is desirous of preserving them in their actual forms, however distorted by false grammar or by obscurities, as specimens of times now passed away, and of religious feelings superseded by others of a different cast. He is anxious also to preserve them on account of the delight they afforded him in his childhood; when the festivities of Christmas Eve were anticipated by many days of preparation, and prolonged through several weeks by repetitions and remembrances.

The version of “God Rest You Merry Gentlemen” that William Sandys included in his Christmas Carols, Ancient and Modern; Including the Most Popular in the West of England, and the Airs to Which They Are Sung of 1833 was notably more archaic in syntax than the version of that song that had been published little more than fifty years earlier in 1775. Many objects, complements, and adjuncts that had previously come after the verb as you would normally expect were moved up so they fell before the verb instead.

Starting with the treasures unearthed by the folklorists, the Victorians set about creating an “old-time” popular conception of Christmas. Others who deliberately used older language constructions during the 19th century include Walter Scott, Charles Dickens, and Washington Irving.

Many new songs were composed during this time with archaic language for the effect of sounding ancient. For example, in 1853, John Mason Neale devised somewhat archaic lyrics for his “Good King Wenceslas”, setting it to the tune of an old Latin song schoolboys of that time would have been familiar with from their Latin classes, and both words and music were included in that year’s Carols for Christmas-tide collection.

So it was in this prevailing environment that John Henry Hopkins in 1857 wrote his “Three Kings of Orient”. In doing so, he deliberately chose these older syntactic patterns of English from earlier centuries. This practice merely reflects what so many other writers were similarly doing during those decades.

---

Modern Confusions

You’re missing a lot of subjects of things that are actually there. Those many verb phrases with present or past participle forms do refer to some noun phrase you aren't seeing; they are not orphaned absolutes.

Yes, there's some use of appositives, but a lot of the mystery may be because you aren’t used to seeing a verb’s objects, complements, and adverbial adjuncts preceding their verbs instead of following them. But precede them they do, and often.

You shouldn’t read too much into the old-style punctuation like the French-style punctuation-spacing or how they sometimes liked an especially long dash after a colon or semicolon to indicate greater pause. Neither should you put too much trust in gleaning some special purpose behind the Deliberate Use of Mᴇᴀɴɪɴɢꜰᴜʟ Cᴀᴘɪᴛᴀʟɪᴢᴀᴛɪᴏɴ that it was first printed with.

A Loose Paraphrasal

Completely off the cuff, but why don’t you try reading it more like this:

We are three kings who have come with presents from the Orient. We have crossed diverse and distant lands following that one star way over there.

Guide us to your perfect light, O star of wonder, O star of night, O star that is bright with royal beauty and is leading west and is still going.

I bring gold to re-crown a king who was born on the plains of Bethlehem, a king who will never cease reigning over us forever.

I have the finest incense to offer its rightful owner, a nearby deity, one all men are sending prayers of praise: “Worship him God on high!”

My gift is myrrh, whose bitter fragrance brings to mind a future darkness of sorrowing and sighing and bleeding and dying and getting locked away in a stone-cold tomb.

Now see him arise in glory: king and God and sacrifice. Heaven sings “Hallelujah” and Earth replies “Hallelujah”.

See if that pedestrian paraphrasing can help you understand the original.

Basically, wherever something looks missing or out of order in the original, all you have to make sense of it is to drag parts of it around till they fit together the way you’re expecting them to. :)

Poetic Devices

The ordering of syntactic constituents is notoriously flexible in poetry, owing in part to poetic devices like hyperbaton and anastrophe to bring the important word to the fore by rearranging the typical order.

• See for example Longfellow’s “This is the forest primeval...” from the opening line of Evangeline.
• Also see Virgil’s “Arma virumque cano...” (Latin for “Of arms and of men I sing”) from the epic opening of the Aeneid, which the poet wrote for Caesar Augustus himself, who just happened to be reigning over the Levant during the storied time of your Christmas carol’s setting.

We Three Kings of Orient are, Bearing gifts we traverse afar, Field and fountain, Moor and mountain, Following yonder Star.

We are three kings of the orient (We are three oriental kings),

We travel long distances carrying gifts,

(We pass via) fields and fountains, moors and mountains,

Following that star over there.

What confuses me about this is the fact that the adjective "bright" comes after the noun.

It comes at the end of the line in order to rhyme with "Light" and "night".

Poems and song lyrics are not required to follow the normal rules.

It's what my dad (facetiously) called 'poetic licence' when I caught him out in a grammatical error. Chambers Dictionary defines it as 'an allowable departure…for the sake of effect, as frequently occurs in poetry'. The key word is 'allowable'.