2 Placed B between 

Dear Nicojo, since you now have many counter-examples, let me give you a situation where $B$ is finitely generated, in line with your question 2). I am going to adopt your notations with the important caveat that $k$ is a ring which needn't be a field .

Theorem of Artin-Tate Consider the inclusions of rings $k \subset B \subset A$ . Suppose that $k$ is Noetherian, that $A$ is a finitely generated algebra over $k$ and that $A$ is a finitely generated module over B.Then $B$. Then $B$ is a finitely generated algebra over $k$.

You might interpret this as saying that when $B$ is sufficiently close to $A$, finite generation is preserved.

You can find the proof in Atiyah-Macdonald, Proposition 7.8, page 81. From this theorem you can then prove Zariski's result that an extension of fields that is finitely generated as an algebra is actually a finite-dimensional extension (Proposition 7.9 page 82 loc.cit.) and then Hilbert's Nullstellensatz is literally an exercise: exercise 14, page 85 . So this result of Artin-Tate is really basic in commutative algebra and algebraic geometry, not surprisingly if you consider the authors (the Artin here is Emil, Mike's father.)

Dear Nicojo, since you now have many counter-examples, let me give you a situation where $B$ is finitely generated, in line with your question 2). I am going to adopt your notations with the important caveat that $k$ is a ring which needn't be a field .
Theorem of Artin-Tate Consider the inclusions of rings $k \subset B \subset A$ . Suppose that $k$ is Noetherian, that $A$ is a finitely generated algebra over $k$ and that $A$ is a finitely generated module over B.Then $B$ is a finitely generated algebra over $k$.
You might interpret this as saying that when $B$ is sufficiently close to $A$, finite generation is preserved.