I think it's a much more subtle and not so well studied question whether various hearts are equivalent as abelian categories. This is definitely false for $\mathbf{P}^1$ (standard heart has infinitely many simples, but the Kronecker quiver has two), but I don't know about elliptic curve and the tilted hearts.

Some of this is covered in Hartshorne: Theorem II.9.3A (f.g. modules) and Exercise II.9.6 (line bundles).

See also Exercise 4.25 on page 141 in Eisenbud's book on commutative algebra.

One thing which is relevant is that on normal varieties rational functions are regular away from their divisors of poles (see `Structure theorem' in III.8 in Mumford's red book), hence if a rational function is bounded on an open set, this set does not intersect the divisors of poles, and the function is regular.

@user2831784: from the exact sequence of Picard groups for $C \setminus p \subset C$ one gets that if $K_{C \setminus p}$ is trivial, then $K = (2g-2)[p]$. This can not hold for general $p$, as if it does for points $p$ and $q$, then $[p] - [q]$ is $(2g-2)$-torsion in the Jacobian.

@Mickhail Bondarko: I think we know $[M(X)] = [M(Y)]$ in $K_0(Chow) \otimes \mathbf{Q}$.

In general, that is in the singular case, class groups do not admit pull-backs for closed embeddings: a typical problem is when the Weil divisor does not intersect the singular locus properly, e.g. contains the singular locus, one may not be able to move it off to make an intersection of the the correct dimension.

@DanielLoughran: my most naive guess after reading Ari Shnidman's reply is: it's necessary and sufficient to have a degree 5 extension L/k. Is this false? Is it at least necessary?