Size of the ring of functions on open subschemes

This question consists of two related sub-questions.

1. Let $$X$$ be a Noetherian integral affine scheme. Under what assumptions on $$X$$ does every open subscheme of $$X$$ have a Noetherian ring of global sections?
2. Let $$k$$ be a field, $$X$$ be an integral affine scheme of finite type over $$k$$. Under what assumptions on $$X$$ does every open subscheme of $$X$$ have a finitely generated $$k$$-algebra as the ring of global sections? I think $$X$$ factorial should be enough (because if there is anything of codimension$$\geq 2$$ in the complement, we can throw it out by Hartogs, and the complement of a pure codimension $$1$$ subset in a factorial affine scheme is affine).