Mayhem: I've managed 2, 4, 9 and (obviously 11). 24 is obviously possible, and 26 has been solved with only bombers meaning it too must be possible.
The only other ones I suspect may be possible, but can't manage myself, are 3 and 16. 3, I'm lost on but I just have a feeling it can be done, 16, my record is 86%.
Seems like you're giving up on Mayhem rather quickly. I can already add two levels to the list without doing any work, because the work has previously be done already.
Mayhem 15 ("The Fast Food Kitchen"): the same exact solution I presented for its clone, Tricky 10, works unmodified here.
Mayhem 16 ("Just a Minute"): the
solution I presented in the minimal skills thread satisfies the criteria of the challenge
I haven't tried Mayhem 17 ("Stepping Stones") yet but it looks pretty solvable to me if you go the ceiling route. Similarly, Mayhem 19 ("Time to Get Up") looks likely to me (who needs climbers when you can just build to the obstacle to let bombers reach it?).
On the other hand, I've actually tried Mayhem 3 for the past two days without success, so I've become more doubtful on that one.
Following Proxima's suit, here are a list of Mayhem levels I'm fairly certain are impossible for the challenge:
1 (splat start)
10 (splat start)
13 (too few lemmings available)
20 (I think your miner or your basher will be in danger of falling to the bottom part of the level, and then you won't be able to save them since you can't assign climber on them)
21 (splat start)
27 (you need to use at least one digger, but then you have no way to stop the digger)
28 (too few lemmings available)
Some of the levels not listed above may well be impossible as well, just that there are still one or two possibilities I need to work through to be more certain.