How can I earn a professor's attention if I am not his/her student yet?
What's wrong with my actions and speaking?
Nothing's wrong with your actions, although they are not likely to lead to success. As for your speaking, the paragraph you quote is not ideal, but I don't think that's why you are getting turned down. Instead, the problem is that the system is just not set up to provide research mentors for high school students. I regularly turn down requests for high school research supervision, for several reasons:
Like many faculty at research universities, I have more local students who would like to work with me than I have time to supervise. This is particularly true for undergraduate research, and I end up turning away many undergraduates because of a lack of time. The ones I end up working with typically have very strong backgrounds, and I'd be reluctant to turn more of them away to free up time to supervise high school students.
It's much easier to vet local students. If they have never taken any of my classes, I can ask my colleagues about them. By contrast, it's much harder to evaluate high school students. It's already difficult to quickly gauge mathematical talent and background, and that's not even enough: I'd also want to know how hard working someone is, how overcommitted they are with other activities, etc.
I don't think high school research is particularly important, and I question whether it's even worthwhile. For the vast majority of high school students (even ones who are exceptionally talented), spending a few more years studying the mathematics that's already known is at least as worthwhile as doing research. The number of people who have genuinely reached the stage where they should be doing research, but who are not yet prepared to go to college, is minuscule. By contrast, there are lots of people whose main motivation seems to be that doing research at a young age looks impressive, and I'm not eager to encourage this.
Of course not everyone will share these reasons, but I think they are reasonably widespread. To maximize your chances of success, you should keep these issues in mind.
The most straightforward one to address is how to evaluate your background and talents. You need external evidence, such as a letter of recommendation from a faculty member, ideally combined with some feedback and advice. You could try taking an advanced course at a local university, or attending a summer math program for high school students. Another possibility is strong performance on a math contest. (That's less meaningful, since it measures only a limited form of talent and doesn't come with feedback/advice, but participating in a contest is less expensive than taking a course.)
It's worth applying to research programs specifically aimed at high school students, such as RSI. There aren't very many of them, and they don't admit many people, so you may not have any luck with this. If you target individual faculty members, you may have better luck if you choose people who have supervised high school students in the past, since you know they are in principle open to the possibility. (You can sometimes tell this from their web sites or CVs.) Faculty members at schools that don't have particularly strong undergraduates may be more excited by the idea of working with a great high school student, while faculty at Princeton have plenty of top undergraduates to work with.
If your search for a research supervisor works out, that's great. But if it doesn't, you shouldn't worry that it's a negative judgment of you. You're trying to do something the world basically isn't set up to facilitate.
It does not sound as though you are supplying any evidence of your ability beyond your own claims. You need to fix that.
I suggest working on your relationship with the teachers, especially the mathematics and science teachers, at your high school. A letter from one of them saying they have a brilliant student who needs more challenge then they can give might work better.
What grades are you getting? If not straight A, what can you do to change that? The most efficient use of the time in high school is to first learn the material it is supposed to teach, so you already have that before going to university.
If straight A, try asking your mathematics teacher for more difficult assignments. Would one of your teachers be able and willing to grade exercises from the textbooks you have been reading? That would give the best cross-check that you really are understanding the material, as well as giving them objective evidence of your ability that they could use in a letter of recommendation.
A letter from a teacher saying "MKR has correctly completed exercises from each chapter of books X, Y, and Z. Here is a sample of MKR's mathematical proof writing." would be far more impressive than your claim to have read books X, Y, and Z.
As I found Mathematics the most supreme beauty, a true Paradise, I started and committed to Mathematics. [...] As I was keen to have a big impact on Mathematics, [...]
This is quite off-putting. What you say here is essentially that you want to be a great mathematician. Since there is no a priori reason to assume that you are better than many other students the professor you address has, such a statement is a little presumptuous.
I had a consecutive study plan and I have studied ... [name of books].
This is good. You give a more or less objective description of your abilities and show that you actually have done serious work.
What's missing is a description of your actual interests. If you say you are interested in mathematics or real analysis, you essentially say that you have not found something that really intrigues you. As research is largely internally motivated, this is not a good sign. Note that I do not say that you have no special interests, but that what you write in your question and what you state as your approach to a professor says so.
So a letter which should attract the attention of a professional mathematician could be the following:
I am (description + why you have no direct contact person).
I have read the following books.
I am fascinated by (special topic), because (honest reason). In particular, I would like to understand (something you are really interested in).
Can you recommend me further directions for my studies, e.g. textbooks dealing with (whatever)?
If you get a reaction like "read XYZ", and after half a year you pose a question which shows that you have worked through this book, you will probably be taken seriously.