Do any mathematicians working in academia have any comments about how one knows if they'll be able to continually generate new ideas to produce publishable research?
I can remember freaking out about this as I was writing my first paper. It was the only publishable work I had ever done, and I remember thinking to myself "What if this is the only idea I have in me? Or what if, instead of getting easier, generating ideas gets even harder as I have to scrounge around in deeper recesses of my brain?"
Fortunately, generating ideas turns out not to be as intimidating as it sounds. In practice, it's very rare to sit down in a chair and say "I shall now think deep thoughts." Instead, any depth comes as a spin-off from much more mundane activities. You read papers, you idly wonder about things, you come up with questions you care about, you figure out how to investigate them, you grapple with technical obstacles, you study things you hadn't realized you needed to know, you chat with colleagues and ask them questions, you work with collaborators, etc. Each of these activities is pretty natural, and they all feed into each other in a complicated web. At any stage you may come up with or run across new ideas, but they are generated organically rather than being something you have to worry about explicitly.
You can expect that a strong graduate program will bring you to the point where you can do this reliably. Of course some people will be faster or more prolific, some will have more striking or creative ideas, some will work on more important questions, etc. You can still improve many of these factors through practice and mentoring, but at that point the question is a little different. Not whether you can do research, but rather how to reach your full potential as a researcher.
So I'd recommend not worrying about this too much. Doing research is a skill that most undergraduates don't have but that graduate schools can teach. Once you get up to speed, generating ideas doesn't end up being a bottleneck. [In fact, you'll end up having more ideas than you have time or energy to investigate yourself. This lets you suggest some to students to help them get started with research, without worrying that you are giving away a limited resource you need for your own research.]
However, given that the opportunity cost of grad school is high and that I already have a stable, good paying job, I would like to know whether I have the talent and ability to be a math professor before I start.
Doing good research is necessary but not sufficient for getting a job in a research university. It's difficult to assess talent and predict career success, but one way to get a crude approximation is by looking at what happened to past students in your position. When you are admitted to graduate school, you can look up former students of advisors you are considering and see what happened to them. For example, you can find lists of students using the Mathematics Genealogy Project, and then you can search for them on the web. This is certainly not perfect: some advisors don't have many students yet, job markets change over time, some former students are just not representative of your situation (if a potential advisor used to be at a less prestigious school, then placement records from that school are not so relevant), etc., and of course there's always random variance. However, it will give you a crude picture. If the advisors you are considering have had many students who got jobs you would like, then maybe you will too. If very few of them got jobs you would find acceptable, then you are taking a much bigger risk.
