How do we know one-way functions can be iterated? Suppose we have some one-way function $h$. Without further specification of $h$, how can we be sure that we can define another function $g(n) = h(h(n))$?

That is, how do we know the range of $h$ is a subset of the domain of $h$?

Relevance to Cryptography: One-way functions are minimal assumptions for the constructing pseudorandom generators. Thus, one-way functions are fundamental to theoretical cryptography (perhaps not for applied cryptography)....but cryptography all the same.
We cannot - we might need another function to adjust the value to be acceptable. Assuming however that $h$ is a hash - this is the crypto site - then it should accept any size of input in bits; up to a seriously high limit anyway.