What about the bonus problems? I hope you are thinking about those too, they’re very interesting and instructive.
On number 6: Clearly it is easy to find a finitely generated algebra B. Even inside a pretty simple B (which ones have you tried?) you can find subalgebras which are *not* finitely generated. But how do you prove that something is not finitely generated?
One tricky thing about this question is that there is so much room to experiment with! I imagine different people will find rather different examples. But if you have no idea how to get started, think about this: what is a finitely generated algebra B that is reasonably easy for you to think about? Inside there, pick an “easy” infinite set of generators for A. Do you need them all? If so, good! If not, try again. 🙂
On number 7: How do you get started? Any good ideas? Presumably you know about or have looked up Catalan numbers, to see that they arise in *many, many* combinatorial contexts. Can you connect any of those with the algebra we are studying here?