The "tl;dr" answers to your questions:
- Not really
- Quite possibly
- Probably not
What you have at the moment is just an aspiration. You really love algebraic geometry and number theory. But you do not know for sure whether or not you're capable of undertaking research in either field. Attending graduate school would tell you whether or not your aspirations are realistic. In the process, you'd be filling in the necessary background in the field, getting a view of the relevant literature, getting insight in ways of tackling research problems and overcoming obstacles, and practicing writing up your results for publication. It's difficult to learn all that on your own.
I should add that the mere fact that you could enrol in graduate school doesn't necessarily mean that you'd be successful. I don't know the successful PhD completion rates in your country, but they're very low where I live (Australia). Obviously many intelligent, well qualified people start out with high hopes and aspirations just like yours, only to exit early. It could be that they find there's a mismatch between their vision of doing research and the reality.
Several answers here have offered examples of people successfully undertaking independent research. Unfortunately, these examples indicate nothing whatsoever about your chances as an individual: survivorship bias spoils the party yet again.
As for being "ostracized from the math community", the reality is that "independent researcher" is a very broad term. It includes mathematicians with a proven publication record, who haven't been able to secure a university position. But it also includes many misguided and even insane people. So you'd need to work hard to maintain and extend any connections you may have with the mathematics community.
Similarly, getting your work published could be problematic. When it comes to examining and endorsing unsolicited papers from an unknown source, I strongly suspect that many mathematicians just don't have time to filter out the trash and investigate the few good items.
A key point that hasn't been mentioned so far is the problem of access to mathematical research and libraries. The widespread shift from paper to electronic formats restricts this access. Maybe you could still visit your local university library and browse, but much of the material can only be obtained online. If you're not a university employee or postgraduate student, obtaining access may be difficult or impossible.
ETA: off topic for this forum, but you may well find that the pressure and uncertainty of an engineering career far exceeds that of an academic career.