What is the lattice point distribution over binary quadratic forms?

Question

Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.

For simplicity, we keep things only on quadrant I of the Cartesian plane and the middle term to zero for the quadratic form.

The question is: as $x,y \mapsto\infty$ what is the distribution of $S$? More specifically, suppose $|f(x_0,y_0)-f(x_1,y_1)|=k$.

1. Say $k=1$, what is the gap (taxi cab or euclidean) in $(x,y)$ as $x,y\mapsto \infty$?
2. Do they get ever larger?
3. Is there a bound?
4. Can we get a function that approximates this gap for each coordinate separately? or even euclidean distance for total distance?
5. Can anything interesting be said about these gaps up to to some limit $x<X,y<Y$?