Some stuff that will help here: $D_4$ has a cyclic subgroup of order $4$ and all other elements have order $2$ (the subgroup of order $4$ is the one given by the rotations).
So we already have a subgroup of order $4$ which is cyclic.
To find one of order $4$ which is not cyclic, we note that the element of order $2$ in the above mentioned cyclic subgroup is central, so if we let $x$ be this element and $y$ be any other element of order $2$, then $\{e,x,y,xy\}$ is a subgroup of order $4$ which is not cyclic.
The part about the quotients by these two subgroups being isomorphic is automatic, as they both have order $2$, and there is only one group of order $2$.
For the second part, we can take the non-cyclic subgroup from above and the subgroup of that generated by $y$. Since $y$ has order $2$ and is not central, the subgroup generated by $y$ cannot be normal.