When talking about elliptic curve over finite fields in ECC, we often assume that the elliptic curve can be written in the Weierstrass form
$$y^2=x^3+Ax+B, \quad A,B\in \mathbb{F}_q$$.
where $\mathbb{F}_q$ is a field of characteristic not $2$ and $3$. (For example, Schoof's algorithm to compute $\#E(\mathbb{F}_q)$ assumes this. )
In practice, would $A$ and $B$ ever be non-integer coefficients(suppose we are working over $q=p^n$)? That is to say $A=\underbrace{1+...+1}_{\text{sum of }1}$ or might we pick $A,B$ to be other elements from $\mathbb{F}_q$?
How would these algorithms that assume the Weierstrass model, generalize to say elliptic curves over fields of characteristic $2$ or $3$? For example, Koblitz curves are a class of elliptic curves of the form $y^2+xy=x^3+ax+1$ in $\mathbb{F}_{2^m}$?