For some context, I was a math major in university, spent most of my career doing something that most would have called “research”, and have a daughter currently employed in biotech research.
Regarding skipping precalculus: I would strongly recommend not doing this. You should take precalculus before you take calculus. As a potential math major, I spent the time to learn all of high school algebra, trigonometry, and pre-calculus very well. Then I found calculus to be very straightforward (including multi-variate calculus and differential equations). I have consistently heard that students who skip on any of the prerequisites typically struggle with calculus. I then found both high school algebra and calculus to be very useful, and to be skills that I used many, many time both in future classes and on the job. Learning algebra and calculus very well is something that is definitely worth the time and effort. The only students who might possibly skip any of these classes are students who are exceptional in mathematics, and these are specifically the same exact people who are likely to be using these skills many times in the future. I would not recommend skipping any of the prerequisites for calculus, and also would recommend taking calculus at whatever your high school considers to be the normal pace.
Mathematics is an area where what you learn today is based on what you learned last week and last year. What you are going to learn next year is based on what you are learning now, and what you learned last year. This continues at least through calculus. At some point there is some degree of “branching out”, but not for quite a while. It really is worth learning each step thoroughly before you proceed to the next step.
It is unusual to get involved in research as a high school student. My daughter who is working on biotech research right now never did any research in high school (as a senior I think that she took one advanced class where they built robots and ran them up and down the hallways of the high school). I think that she was half way through her bachelor’s degree when she first found opportunities to do research as a university student. I do not recall doing any research at all as a high school student.
You will become a stronger student as you get older. You might want to see how you do in your current chemistry course before you decide when to take AP chemistry. You will want to take other sciences in high school. One option for example might be to take either biology or physics next year, and then take AP Chemistry your junior year of high school. You might want to discuss this with your Chemistry teacher closer to the end of the year (after you and the teacher both see how you are doing this year).
Most of us don’t. At a few highly ranked universities (MIT and Caltech come to mind) students do study quite a bit. There are very good research opportunities at a wide range of universities.
However, research, and life in general, are much closer to being a marathon than a sprint. We need to take things one day at a time. Someone once said “Make haste, slowly”. The point is that you cannot do everything today. You do what you can today, rest up a bit, take time to have some fun, then you do what you can again tomorrow.
Also, you need to take the time to learn the basics before you will be ready to do more advanced work. This takes time. You will get there, but not right away.