Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order.

$\textbf{Questions:}$ $\newcommand{\Spec}{\textrm{Spec }}$ $\newcommand{\cO}{\mathcal{O}}$

1.) Is the natural map $\phi: \Spec \mathcal{O}_K \rightarrow \Spec \mathcal{O}$ flat?

2.) How many distinct primes (can) lie under a given prime? All but finitely many local rings of $\cO$ are canonically identified with local rings of $\cO_K$. Since it is not obvious to me that $\phi$ is surjective, how many more primes are in $\cO$ than in $\cO_K$ (w.r.t. the canonical identification above).

3.) In $\mathcal{O}_K$, and dedekind domains, prime ideals can be generated by two elements. How many elements are required to generate prime ideals of $\mathcal{O}$ ? Is it possible to give an answer depending on the degree $n = [K:\mathbb{Q}]$ and the index $[\mathcal{O}_K: \mathcal{O}]$?

4.) Is every ideal $I$ of $\cO$ also a proper $\cO$-ideal as is the case for the maximal order? That is, has ring of multipliers $R$ exactly $\cO$. (The rings of multipliers $R\subset K$ is the subring of elements $\alpha$ so that $\alpha \cdot I \subset I$). Certainly $\cO \subset R$.

$\textbf{Note:}$ If need be, feel free to assume that $K$ is quadratic imaginary. I'm primarily interested in this case, but I would like to have a clearer picture of the general situation.