I've run this adventure several times, and the group has never done it by actually jumping across the disks. Jumping across the disks is obviously absurdly dangerous, and high-level adventurers generally have at least one alternative movement mode among the group.
The way this room has always worked (in my games) is that the group finds a way to cross the chasm, but the method they've found doesn't work for the whole party at once. Maybe someone uses a fly spell on a strong character that can carry one other character with them, or someone polymorphs or wild shapes into a giant eagle, or summons a flying monster, or casts dimension door, or has the Athlete feat, or they just cast spider climb.
Then some of the group cross the chasm, and they're like "okay, now the half of us that have crossed the chasm will wait in this cave, and the eagle will fly back to the start and pick up the rest of you --" and that's when Ctenmiir attacks. It's a wonderfully dramatic scene, but it's also kind of a bad experience for whatever group members get left out of the first few rounds of the combat, so it's not my favorite part of White Plume Mountain.
I don't run the crossing as a combat, so there aren't explicitly any rounds to keep track of; I just periodically tell them that a geyser is exploding.
You've asked about mechanics for jumping from chain to chain. The 5e version of the adventure mysteriously omits the rules for this, but it might interest you to know that the 3.5e version has a table with DCs. In that edition, it is DC14 to jump cleanly from one disk to another; a result of 10-11 or 12-13 gives the adventurer a chance to save themselves by making additional checks, and a result of 9 or fewer means they have fallen into the boiling mud. This edition also discusses what to do if the adventurers tie themselves to ropes while crossing (a good idea, but not one that any of my group has tried).
The DC15 STR check is for "climbing the chain" or climbing the walls; it's not clear why anyone would want to climb a chain, so that part of the rule seems sort of pointless. If someone on a disk is hit by a geyser, there's a Strength saving throw to not be knocked off of a disk.
You've asked about whether the disks are on an incline, with each disk higher than the previous one. I don't think they are. The only indication that the disks might be on an incline is the image in the 5e version of the adventure; the 5e text and the 3.5e text make no mention of this, the image that accompanies the 3.5e version shows the disks as not being on an incline, and I don't think it's realistic to have the boiling mud in the floor of the cavern be anything but horizontal.
You've asked about falling damage. The 5e version omits the rules for this; the 3.5e version says that the falling damage is only "1d6 lethal damage and 2d3 nonlethal damage, just like falling into water". 5e has no rules for falling into water, so as the DM it's up to you how to rule that.
You've asked what happens if someone falls. If someone falls, it would be up to the rest of the group to fish them out before they die, perhaps by dropping a rope for them to grab on to. I suppose that, if they were immune to fire, they could instead swim to one of the ends of the cavern and try to climb the rock.