I'm having a go :D
The river to the underworld
Styx (or Stux) was the river that needed to be crossed to enter the underworld. Charon was the ferryman.
Gaia’s daughter all unfurled
Hmmm. Is this a crossword reference to an anagram? Gaia's family line is complicated but there were 7 female Titans that she bore:
QuoteDIONE The Titanis-goddess of the Oracle of Dodona. She was the mother of Aphrodite by Zeus.
MNEMOSYNE The elder Titanis-goddess of goddess of memory, words and language. She was the mother by Zeus of the nine Mousai (Muses).
PHOIBE (Phoebe) The elder Titanis-goddess of intellect and the Oracle of Delphoi (Delphi). She was the grandmother of the god Apollon.
RHEIA (Rhea) The Queen of the Titanes (Titans) and goddess of female fertility and the wild mountains. She rescued her infant son Zeus from the maw of Kronos (Cronus) who had devoured her other five children.
TETHYS The elder Titanis-goddess of nursing and the underground sources of fresh-water. She was the mother of the Rivers and Springs.
THEIA The elder Titanis-goddess of sight and the precious value of gold, silver and gemstones. She was the mother of Sun, Moon and Dawn.
THEMIS The elder Titanis-goddess of natural order, divine law and tradition, and the oracles of the earth. She was the mother by Zeus of the Fates and Seasons.
There were also younger female Titans one of which was Styx. So maybe the lines in the clue should be taken as a pair :g:
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Fourth minor bear in vain
By viper’s sting was slain
Ursa Minor's (little bear) fourth brightest star is now known to be Epsilon but it would originally have been Delta
Also known as Nemesis
The doom of Chimera
She is tasked with soothing pain
Suckling Odysseus from afar
Mother of the mother of
Leader of the Titans
He transformed into a hawk
A daughter of Daedalion’s
The piercer was how he was known
Mother of Ulysses
Ruler of the winds
God of night, primordial flees
Zeus’ namesake now lies in Sol
Achilles’ favoured horse
Women of vengeance infernal
The vain queen rides not forth
A final word, a course to follow, a poor miser’s sum
If you would understand it all, seek Fibonacci’s Zephyrum