The Inns in The Great Hunt offer somewhat of a 'regression' compared to those found in The Eye of the World and it is one more way in which it shows the second book of the series was actually the first Robert Jordan completed.
While the Inns in the first book were full of myth and legends references, those in The Great Hunt have much simpler symbolism and often relate mostly to plot elements.
The first we encounter, Easing the Badger in Illian, is some sort of an inside joke. Probably a reference to an obscure folktale (real or invented by Jordan), the sign shows a man with a silver shovel dancing with a badger. The Inn has a rough clientele, but the innkeeper Nieda keeps all these badgers happy, offering sailors and the like drinks and, later on when the mood in Illian won't make drinking enough to 'ease the badger', lewd entertainment. Badgers are, to say the least, ill tempered. It may be a reference to sailors forced to go to the sea for long and caught like a badger in its hole. The Inn sign promises that at this establishment your 'badger' will be eased - with booze and innuendo.
At the second degree, this is reflected in the events that take place there. Bayle Domon in the The Great Hunt episode will take the decision to 'ease his badger' at this Inn, finding a way to escape the Darkfriends pursuing him since Saldaea. In The Dragon Reborn when we return to Nieda's Inn, both Faile and Perrin will be in a badger-like temper, especially Faile (the fish hater) who growls, sulks or complains during the whole meal. At this location, Perrin's group will also manage to ease their badger by confronting and defeating the Grey Men that Ishamael had tracking them and Rand since the Mountains of Mist... only to gain a new pursuer in Sammael.
With The Nine Rings, in the village of Tremonsien, Cairhien, Jordan made his first (but not last) explicit Lord of the Rings reference. Here it refers to an in-world story Rand loved a lot growing up. Of course, we have to understand RJ loved the book as well. I can't say I'm a huge fan of this sort of too blatant references to the real world (outside all the mythological/legendary stuff, I mean). It tends to throw me out of the secondary world - but thankfully Jordan didn't overdo it (the Mr. Underhill reference in a recent book was more cringe-inducing). As for the reference here, it obviously refers to the Nine Kings of Men in Lord of the Rings who were seduced by power and glory and fell into Sauron's grasp, with his One Ring that binds them all. The parallel in The Wheel of Time and this book is of course to Lanfear, who tries to entice the heart of Rand, to light a desire for immense power and glory in him, to make him fall for her, and to the Shadow. Located near Kinslayer's Dagger and the location of the male Choedan Kal, The Nine Rings may allude not only to the desperate hubris of Lews Therin and that of the makers of the Choedan Kal - an artefact with enough power to destroy the world - but also to the vainer hubris of the Kings of Men, here Galldrian, who tries to unearth the sa'angreal, and his predecessor Laman who cut the Tree. Finally, this Inn is visited right after Rand stole back from the Gollum figure of the story, Fain, his 'precious' without which he cannot feel complete.
In Cairhien itself, The Defender of the Dragonwall shows a King, probably Galldrian (it's doubtful any Inn would dare celebrate Laman at all, even less when the current King's House is a big rival of the Damodreds) triumphing over an Aiel. He stands over an Aielman, his sword at his throat. This is primarily a reference to the events at the Inn: shortly before Perrin and Verin have met Aiel, and we got even more clues about Rand's true ancestry. At this Inn, Rand gets nearly overwhelmed and embroiled in the Game of Houses, in the end receiving invitations from both Barthanes Damodred and Galldrian. Funnily, all this ends not with the Aielman being killed, but with "The Defender of the Dragonwall" himself (it's a title of the monarch of Cairhien) being assassinated, having ruthlessly played the Game against the wrong man, Thom Merrilin, in his efforts to get to Rand. The Inn itself is burned out by Fain, in a variation of the 'pattern' of events of Winternight: the chest hidden upstairs (ie: Tam's in EOTW) doesn't contain a sword this time, but the Horn of Valere. Rand isn't threatened (the moonlight risks making him seen) and helped (hidden by the moonshadows) by the Full Moon that seems ready fall on him (as it is described in The Eye of the World), he is stalked by the the Moon herself, Daughter of the Night, helping and threatening him at the same time. Where Fain caused an explosion, seemingly as a diversion, at Winternight, here he causes a fire for the same motive. Thom and Rand discuss Aes Sedai and Aes Sedai help, as they did on the morning of Bel Tine. The dying man isn't Tam but an older man who defers to Rand: Hurin, on whom Rand must rely as a guide. Desperate, he isn't met by the blacksmith Haral Luhhan but by the blacksmith apprentice. Again, the Wisdom can't save the dying man, and it's the arrival of an Aes Sedai that save Hurin... again with the feeling there's a hook involved.
RJ liked to have his plots progress using repeating patterns like this, mixing up similar ingredients in different ways; while most of these 'repetitions' are too subtle to catch on without a few re reads (and even sometime with analysis of the elements), they still create the subconscious impression of a Pattern on readers, which no doubt was RJ's goal using this technique. On a side note, the Inn sign is very possibly inspired by the classic depiction (used in variants on tons of paintings, famous and less famous) of the triumphant Archangel St.Michael and the beast (the Aiel are seen little short of beasts and demons-darkfriends to some) - and to various Dragon slayers.
The Great Tree, an in-world allusion to Avendoraldera Laman had cut down, is obviously foreshadowing of Rand's destiny and the developments to come in The Shadow Rising. In many ways, the Tree of Life represents Creation/humanity needs to save. As the Tree of Man, Avendesora is also a representation of Rand himself - Rand as the Great Tree, solid but which if can't bend enough could be broken by the 'great gathring storm' ahead. More immediately, 'The Great Tree' also mark a turning point: from there, Rand will depart for Almoth and Toman Head, which a verse of the Dark Prophecy and Verin's explanation linked to the Tree of Life. This is also from The Great Tree that Rand will visit the site of the ancient Grove (ie: Barthanes's palace and the Waygate) and it's also from The Great Tree he will depart to visit the Great Trees, those of The Stedding.
The Bunch of Grapes alludes to a metaphor Jordan liked. As he often did, the metaphor isn't explained in this book but in another, The Shadow Rising. Grapes can't do nothing to stop being eaten one by one or sent to the press, and the winemaker doesn't think twice of the bunch of grapes he sends to the press to make wine. This is a very apt reference for the Foregaters, mostly peasants who left their farms after the Aiel War and would soon die or become homeless refugees of the civil war, after the Foregate burned, victims of a King and nobles who care only for the wine and don't spare a thought for the grapes they need to send to the press. More immediately, it's Thom's lover Dena that gets sent to the press to make wine, assassinated by one of the King's thugs.
To accompany this Read-through post, were are publishing the The Great Hunt entry of Dew Drop Inn: Wheel of Time Accommodation, where you'll find the complete listing of the Inns from the book, a few illustrations of Inn signs and complementary and alternative interpretations of the Inn names and signs.