Your iterate function as written relies on a top-level define feature, which pure lambda calculus lacks (motivating the use of fixed-point combinators).
No, iterate is just a helper function to convert a Haskell integer to a Church numeral. If the inputs were directly represented as Church numerals, it wouldn't be needed and you'd just replace every instance of iterate n with n itself. I thought this would be evident to someone who had read the article and understood Church numerals, so I didn't go into detail about it.
