Water Rondo HAS to be gotten in the first half of the game before going to the floating island. While in Narshe theres a scene involving Lone Wolf and Mog. You have the choice of saving the golden hairpin or mog. Save Mog. You can get more pins later but if you don't save Mog here he won't join up with you. Afterwards go to the Veldt with Mog in your party and go down the serpent trench.
Also the only way to get the other place specific dance ( I don't remember the name) you have to bring Mog to Narshe in the world of Ruin.
To get Umaro: In the world of Ruin go to Narshe. There's a scene at the top of the mines where the Esper Tritoch will be (Yes it's the same place as in the World of Balance). After fighting Tritoch the ledge will crumble away. Choose to jump down and follow through the cavern inside. Umaro will be found near the end and if Mog is in your party he will join the team.
I would mention Gogo but I don't remember the name of the island the Zone Eater is on.
