Coefficients of Elliptic Curve over Finite Fields
$A$ and $B$ can be non-integers if you're working with an extension field. For example, this happens with pairing-friendly curves where you need to work with elliptic curves over e.g. $\mathbb{F}_{p^2}$.
Depends on the algorithm. For counting elliptic curve points, the "hard" part is computing $\#E(\mathbb{F}_p)$. If you compute that, than it's easy to compute $\#E(\mathbb{F}_{p^k})$; see this answer. In the Koblitz curve case it's even easier since computing $\#E(\mathbb{F}_{2})$ is trivial (it's either 2 or 4 depending on the curve).