Weber's class number problem and real quadratic fields of class number one

This statement is most certainly nonsense; I guess what she meant to write was that in order to prove the existence of infinitely many number fields with class number $1$ it is sufficient to prove $h(F_n) = 1$ for infinitely many (and therefore for all) $n$.

Let $K$ be any subextension of $F_n$. Then the natural map of class groups $\text{Cl}(K) \rightarrow \text{Cl}(F_n)$ is an injection. Indeed, you can see it in terms of unramified abelian extension via global class field theory. If $L$ is an unramified abelian extension of $K$, since $F_n$ is totally ramified at primes over $2$ in $K$, we have $L \cap F_n = K$. Therefore the compositum $L \cdot F_n$ has Galois group $\text{Gal}(L/K)$ over $F_n$.