Have I missed or misinterpreted anything? Thanks!
We start out in the queue in some sort of barn or what have you, more or less corresponding to the "real world" in the film. The queue and loading area are a transition into the animated world of Br'er Rabbit and company, where we begin the ride by floating around the briar patch and various character's homes. The visible mountain itself is Chickapin Hill, the home of Br'er Fox, who's determined to catch Br'er Rabbit, often in spite of the stupidity of Br'er Bear.
Eventually, we slide down Slippin' Falls (the first drop) and end up in the bog, where things are generally cheerful in nature. Br'er Rabbit is leaving his home in the briar patch to find his laughing place. Amidst various miscellaneous characters from America Sings, we see Br'er Bear dangling under a tree; Br'er Rabbit has tricked into taking his place in Br'er Fox's trap. At some point, he gets away, and wants to join Br'er Rabbit in his laughing place, but Br'er Rabbit tricks him again and instead points him in the direction of a number of beehives. We drop down and down, past the bees, and into the laughing place.
Naturally, things are even cheerier here. Everyone's singing and laughing except Br'er Bear, who's got a noseful of bees to deal with. Generally strange displays surround us - these have no basis in the film, nor do many of the characters - and everything seems to be going great until Br'er Fox manages to snare Br'er Rabbit by causing him to lash out at a frustratingly silent doll made of honey. (Tar was deemed too racially charged due to its dark color.) In their burrows deep inside Chickapin Hill, the rabbits and opossums lament Br'er Rabbit's fate.
It's off to Br'er Fox's home in the hill, where the trapped Br'er Rabbit is going to be cooked for dinner...until he cleverly uses reverse psychology by pleading with Br'er Fox not to throw him back into the briar patch. The ruse is successful, and we experience Br'er Rabbit's fortuitous fall back into his thorny home. After floating through the outdoors a little more, much like the start of the attraction, we see everyone celebrating Br'er Rabbit's return, as well as the antagonists' fate in the jaws of a hungry gator.
Oh yeah...and there are vultures.