Normal amount. Sliding off is generally preventable with stuff fastened at each end. But of course, he is then trapped in his little skybag. Rods are MUCH more useful if you have two, one in each hand.
1)
Unless he has a bungee cord attached to the rod, he is getting stopped hard by the rod. Roughly as hard as if he had hit the ground. Possibly even harder, since his legs can't make contact with the ground first and spread the impulse on his precious internal organs across a longer period of time.
Mind you, he could arguably "pulse" the rod and fall a few feat at a time. That might work. It would take a while to get down, though.
2)
With any degree of ingenuity, it's a nonissue. Sissyl's approach is one. As an alternative, a bag around the rod can be used in a number of ways to also solve it. The only question is whether a material can stand up to a sudden stop under the weight of a human(oid) being, but frankly you are already ignoring that for the rope - a guesstimate puts the max force on the rope when stopping an object from a 30ft fall as 100 times higher than just holding something suspended in the air (and I suspect that number is a serious underestimate).
There aren't really any rules for this. First off, the game doesn't have a climbing harness in it, it has a climber's kit, but that isn't particularly the same thing.
Forgetting about the immovable rod for a bit, if we think about a real world method of climbing, you would attach a piton to secure your rope, climb above it, say 10' and then you might fall, falling a total distance of 20'. This is an absolutely reasonable thing to imagine, but the rules don't go into anywhere near this level of detail. The climb kit just gives a +2 bonus, and if you fail your check by 4 or more you fall, presumably all the way down.
I would be quite reasonable to house-rule this sort of situation though.
Going back to your original example, I would probably have them take the damage for how far they have fallen (if they don't activate the rod in the first round as a second move action after they jump off it will be max falling damage, and they will have already fallen about 500') but I would probably subtract 1 from every die of falling damage (min 1) to represent being stopped by a harness is quite a bit more forgiving than hitting ground. This would be a house-rule/GM call, not anything official though.
As far as the rope sliding off thing, anyone competent with knots (which would include anyone with the climb skill, at a minimum, since use rope isn't a thing anymore) should be able to secure a rope to a stick in such a way that it would be unlikely to come off. Beyond that, there is no particular reason one couldn't quite easily come up with other methods such as attaching hardpoints to fasten your rope to.
A character cannot cast a spell while falling, unless the fall is greater than 500 feet or the spell is an immediate action, such as feather fall. Casting a spell while falling requires a concentration check with a DC equal to 20 + the spell's level. Casting teleport or a similar spell while falling does not end your momentum, it just changes your location, meaning that you still take falling damage, even if you arrive atop a solid surface.
The quickest spell you can cast that is not an immediate action is a move action.
Given that max damage (20d6) occurs at 200', and that you need 500' to cast a spell, I don't think you have time to make the move action to activate the rod before you go splat at any distance less than 500'.
So any distance you can activate the rod, you are already at max damage level.
1) How much fall damage would he incur? The normal amount, or does being caught by the harness reduce it in some way? If so, by how much?
2) What check/roll would be used to determine if the rope/carabiner/whatever slides off one end of the rod? As far as I know, there's nothing on the ends of an immovable rod that would stop them from sliding off. If the PC didn't freeze the rod in place exactly level, he'd slide off one end, wouldn't he? Even if it was level, swinging or other lateral movement could cause it to slide one way or the other.
1) 20d6 each time
2) It is a reflex save to catch yourself falling, so I would say it is a reflex save to not slip off the rod when you activate it. You could also argue for a Fort save to hold on, but that has less rules support. Once activated, I think a Climb Check DC would be appropriate to have yourself securely tied off as you descend./cevah