Taxing 16 "The Arena" - 1 turner, 1 bomber, 1 builder
maybe? (I didn't solve this - but the record is lose-2, and I'm pretty sure this level can't be done without the turner and builder, meaning 1 bomber?)
Taxing 17 "T Minus 5 And Counting" -
None (the 100% solution uses only one skill, a basher; the same route can be done, lose-1, with a bomber replacing the basher)
Taxing 18 "Snake" - 1 blocker, 16 turners, 3 bombers (the max-saved solution achieves 3 bombers; however there aren't enough spare bombers to use this trick with the otherwise-normal solution and save a turner)
Solving it without the floater is done by building from immediately below the start (at the edge of the block is okay), sending a climber out of the containment area, mining into the left wall then immediately mining after turning around - this should let him fall to the yellow block, which he then bombs on to create a safe fall to the bottom. Release the crowd with another bomber, and build-mine through the final wall.
Eliminating all the other skills is fairly trivial.
Turn to the right when under the rope, build, build back on top of the wall (make sure to drop onto it!), build twice towards the exit. This solves the level with a turner and 4 builders.
You can then solve it with 3 builders by using the deflector block near the bridge together with a turner, building from under the entrance, blocking, and building over the wall, the rest should be obvious from there, though note that the 2nd builder must be assigned very precisely so it starts from the top step of the 1st builder's bridge (you've only got one frame to do this, but if you carefully position the mouse and rapid-fire click, you'll get it fairly easily).
I'm 100% sure that it's not possible to solve without the turner, though I'm not completely convinced that 2 builders isn't possible - although I can't see anything.
Regarding Mayhem 1 - I
think it's possible to eliminate some skills, even though the normal way uses everything. However, in both cases, the timing required is insane, so I'm probably not going to be able to pull it off myself. I can't say for sure these are possible.
To eliminate 1 climber - Using release rate manipulation, set it up so that the bomber used between the two blocks on the far side, is timed so that one lemming slips past and can dig, without him having to get ahead by being a climber. If it helps, the digger that builds across the gap, can dig one slice further than usual, which creates an area that can hold the crowd while still successfully crossing the gap.
To eliminate 1 bomber - Don't bomb the lemming that digs away the first block. Instead, use a "Fill The Floor" approach (alternate RR99 and min-RR, to try and group up lemmings even more tightly than RR99 alone does) to slip the crowd past while he's digging. Note that the tile at the bottom of this column is water, so this cannot be used to make a save 79 solution, even though the arch tile next to the column is solid. For the same reason, this cannot be used to save an (additional) climber.