Asking a faculty member at my university that I have not met(!) about his research, and about courses that deal with his specialty/my career goal?

It is completely fine to send an email. Your request is totally reasonable. You are a student, at university to learn. The professor is at university to teach. If they get offended by a simple enquiry that shows interest in learning more about their favourite subject then you have a bigger problem on your hands.

It will not be the first or even the hundredth time this professor has received an email from a student they don't know and writing such an email is not going to cause offence in any way, shape or form. The worst that can happen is you get a polite "Sorry, I am too busy" or perhaps no reply. The professor will not be offended by you asking -- in fact, it's more likely they will be annoyed or frustrated that they don't have the time to help you, not at you for asking.

Finally, I would not start off with such a grovelling/self-deprecating tone. There's no need to apologise for sending a perfectly innocuous email. However, please bear in mind that academic cultures do differ, and I'm writing from a UK/Western European perspective, where the culture can be a little more blunt and far less deferential towards authority (i.e. professors) than in the USA (where I assume you are from). To me it would come across as more professional if you write to them as though you are a peer wanting to initiate a scientific discussion, rather than as a lowly student who isn't worthy to interact with an esteemed professor.

In summary, the best thing to do is make sure your email is short, polite and to the point. Briefly introduce yourself, ask your question and thank them in advance for their time.

This is coming from a US perspective, so take that as you will.

Professors are pretty busy and generally just want to cut to the chase. A good email is one that is written clearly and concisely enough that it can be scanned in a few seconds and the gist of what you want to say is clear. Often after a quick scan the professor will decide if they are interested and want to reread it more carefully.

That being said:

• Introduce yourself briefly: Name, undergraduate, major, interested in X
• State what you want clearly and briefly. One bite-size paragraph for each key idea (~3 sentences each) has worked well for me in the past.
• If you want you can close with something like "given your expertise in this area I would love to get your input, but I understand if you are pressed for time".
  • Since it is a given that professors are incredibly busy, however, I might consider leaving this off. Some people don't like it when you state the obvious, while others might view it as a courtesy, it's really your call.
• Showing a bit of passion can be good. If fin. tech is really your 'dream' job then try to convey that in some way (just saying, "its my dream to work in x" is good enough). Professors are often very passionate about their work and like to help students that share some of that passion too.

I also think you should consider whether you want to ask to meet with them briefly rather than discuss things over email. Many professors are more than happy to share their wisdom and expertise with students. If you are more comfortable with email though then that is certainly fine.

If you do want to meet, then in the 'state what you want section' mention you want to pursue mathematical finance, that there is no apparent program at the university, that you are looking for guidance on what to study, and ask if they would be interested in a brief conversation about it sometime.

Don't be worried about these sorts of emails having a 'bad reputation'. The kind of emails that actually have a bad reputation are low-effort correspondence, often from lazy or entitled students. That is not the case at all here, your university is not setup to provide what you want and this professor is uniquely positioned to be able to help you. At worst he will read it, be too busy, and just move on without a second thought.

What to Avoid

Don't grovel like you did in your proposed intro, apologizing for any perceived offenses your email might be about to commit. This doesn't help the professor know who you are, what you want, or why he should keep reading the email and will likely come off as rather irritating. The professor fully understands the situational dynamics, so there is no need to make them explicit. Just stick to the facts. (quantitative people especially tend to like this)

As long as you are polite, direct, and respectful you should be fine.

Don’t over think it I would drop a brief email saying as you wrote above I’m @simonshampoo studying Math and CS and have questions a, b , and c and was wondering if I could pop in during your office hours (or your university’s equivalent), or some other convenient time, to discuss.

You don’t need to be apologetic (as @astronat ‘s answer rightly notes ). Few faculty are going to decline to see serious students with interest in their area. I get such requests from time to time, mostly from maths majors, and have always found them interesting conversations. Many (most?) faculty, dislike answering questions which are available via google, but are often pleased to answer those in their area which are not.