Comment by penteract
I think you've got the first claim wrong - for a church numeral N, (N b g f) appears to give the composition of g and f with f taking N arguments.
2 b = (λf x.f (f x)) b
~> λx.b(b x)
= λx.(λgfy.(g (f y))) (λabc.(a (b c)) x)
~> λx.(λfy.(λabc.(a (b c)) x) (f y))
~> λxfy.(λbc.(x (b c))) (f y)
~> λxfy.(λc.(x ((f y) c)))
= λxfyc.(x (f y c))
(3 b) ~> λihgfx.i (h g f x)
Ah yes, you're right. I messed up the associativity in the reductions.
It still does what most interpretations would consider the "nth composition combinator":