Is paradox recognition paradoxical or not paradoxical?
Russel's paradox does not apply to the lambda calculus. Intuitively speaking, Russel's paradox comes up when there is a level system (e.g. sets that are members of “bigger” sets one level up), and there is a form of negation. The lambda calculus has neither: all lambda terms are at the same level and there is no negation.

I don't know what you think you mean by “The lambda-expressions that may not be applied to themselves”. Any lambda term can be applied to any lambda term.

An analog of Russel's paradox (or rather the similar Burali-Forti paradox) arises in the higher-order typed lambda calculus, where terms have types that themselves have types. If $\mathsf{Type} : \mathsf{Type}$, it's possible to construct a well-typed term that of type $\mathsf{Type}$ that does not terminate. See Example of a false proposition when assuming Type : Type. As can be done in set theory, type theory can solve the problem by adding a universe stratification: base types have the sort $\mathsf{Type}_0$ and higher-order types obey $\mathsf{Type}_n : \mathsf{Type}_{n+1}$.