I think the entire discussion basically revolves around this:
The Oh-Noer I think should cancel out the other permanent skills.
That's precisely what came to my mind when I thought about what the suggestions made in this thread would imply. But that also means we need to think about whether this applies to all permanent skills we have so far.
1.) An oh-noing disarmer should not be able to disarm traps anymore, although he can be eaten by them. (A oh-noing climber also stops climbing for example)
Indeed, Oh-Noers can be killed by traps - I've found out by accident on my Lemmicks level "The Eye of the Needle", where I wanted to stone right between a teleporter and one of those Space "shooting bird" traps. Even though I lost the Stoner from the skill panel due to assigning it, the lemming still got shot by the trap rather than turning to stone.
Since Oh-Noers can only get onto a trap trigger by falling, and falling into a trap kills Disarmers, it makes sense for oh-noing Disarmers to die to traps as well. However, this doesn't necessarily imply the Disarmer loses the skill due to oh-noing, because he would also die if he simply fell onto the trap trigger regularly.
Floaters and Gliders can't oh-no, and coincidentally, Oh-Noers can neither float nor glide, because they never reach splat height anyway before exploding, as I've repeatedly outlined in this thread. So we never actually see whether they "lose" the skill, unless you'd want to argue that floating or gliding lemmings already open up their parachutes very early during a drop, i.e. even before a regular lemming would reach splat height - and Oh-Noers don't.
With Climbers, there's an important difference to what you've outlined here:
The Climber stops climbing, yes, but it doesn't "lose" the Climber skill in the sense that it doesn't suddenly fall off the wall. That's actually what Shimmiers used to do when they were assigned Bombers, and then we changed it so that Shimmiers also explode instantly. It was the only case where I had to slighlty adapt a level to changed Shimmier behaviour, but I was happy about that, because any lemming without ground under his feet should explode instantly. In the Climber's case, one could argue that his feet are pointing towards the wall, hence he oh-noes, but Shimmiers don't. Consequently, if we were to introduce the Magno-Booter from Lemmings 2, that lemming should oh-no when turned into a Bomber, even if he's currently dangling from the ceiling. (L2 circumvents this problem by not enabling the player to assign skills to upside-down Magno-Booters at all, but I think that makes the skill very limited in its application.)
Also, on a purely visual level: Athletes that are assigned a Bomber or Stoner don't suddenly return to having green hair and blue clothes when they oh-no...
Instead, with the Climber remaining in its place when oh-noing and sort-of "standing in the air" while doing it, one could argue that Swimmers should do the same, i.e. "stand" on top of the water area an finish the oh-no-animation plus the explosion, rather than continuing to fall.