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.