How can there be an exit that Climbers can't reach, but non-Climbers can? Same question for a skill that "needs to be given to non-Climbers" - why can't it be given to the Climbers as well?
I can one up a theoretical answer here, and show some actual examples from my levels.
For the former - "an exit that climbers cannot reach". Although not
directly used to prevent reaching an exit here, the overall goal remains the same - the climber cannot turn around at either side of the level (without wasting a skill to do so, one that the player cannot actually afford to waste - this is the counterpoint to the "why not just use another skill?", as that's just like any other situation where a skill can be used to perform a backroute and doesn't invalidate the overall setup in general). You could apply this specifically to the exit - have a wall like this, or multiple walls like this if you like, with the exit at the bottom. Really, this just comes down to the very same "climbers may take a different path than non-climbers" that we're already very used to, and just looking at it from a different angle - "what if the path the climber would take is the one we DON'T want to use, instead of the one we do?"
For the latter - "a skill that needs to be given to non-climbers, and can't be given to climbers as well". This level isn't
quite exactly that situation, as the climbers / floaters are preassigned, but it illustrates the general idea of how one would acheive this. You need to save
all 20 lemmings 38 of 40 lemmings. Were they not preassigned, you'd have climbers for half the lemmings and floaters for the other half - you couldn't send all
20 38~40 to the same exit, because there aren't enough climbers to send everyone to the right, and there aren't enough floaters to send everyone to the left. Of course in another level, this could simply be "you need one climber and one floater, each on a different lemming", not necesserially "every lemming must be one or the other"; the reason it couldn't be given to the climber simply being in general "because there aren't enough left after giving them to everyone else who needs one".
OK, sure - nicely done. Maybe I should have said directly in front of instead 
Here's a scenario which addresses all of the above:
Keep in mind also that, as shown by the examples above, in real-world levels there are stricter limitations (and the level author can add more of / alter these as needed, generally) on what the player can actually do to bypass it, as opposed to theoretical scenarios where infinite skills of every type are available, as are infinite useable (and loseable) lemmings and infinite time if it comes to that. Some of the terrain involved may be steel or one-way, or a needed skill may not be available (either not in the skillset entirely, or needed elsewhere in the level / provided as a pickup that would not be reached using this approach).
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