If I am interested in a particular subject/club, if I reach out to the professor could it help with my admission to the school?
I’m a teacher at a private school. Reaching out to me would be of absolutely no use to you.
Professors teach in colleges/universities, and not all college instructors are professors. At the HS level, they are teachers or instructors.
But to answer the question, no, it won’t help.
I’m going to disagree here. It CAN help. Most of the time, it won’t help. If you know someone who is a teacher at a prep school, and you want information and help in applying there, some teachers, staff there might help you in that regard. Some may simply show you to the admissions office and give you the stock materials there. Some might reach out a bit more and give you some time and specific information, help you out a bit. Rarely, but sometimes, if you find someone at a school that does have some influence over admissions and the person decides you are a particularly stellar candidate, it could give you a leg up. But as the answers above clearly state, that is not something you can count on happening. The average teacher at a school, professor at a university, does not go around advocating for admissions of kids they happen across or who ask them questions about the school. Most are happy to give you the time to give you info about the school and talk to you, answer questions, but they cannot and should not be the conduit for admissions. The chaos that would ensue, would be horrific if you had any and every connection to a school involved in championing admissions candidates. That is why there is an Admissions Office.
If you are an internationally competitive Math whiz, reaching out to Exeter’s Math department might help. Otherwise, it generally won’t affect admissions unless you reach out and embarass yourself horribly or make a terrible impression.
Does the same apply for sports and meeting coaches?
Sports depends on the school. At some schools, certain sports are so highly valued that coaches can have incredible sway. That is true of the arts at some schools, too, where applicants are encouraged to submit portfolios for evaluation or even participate in auditions. It really varies by school. It also depends on what the school needs. If their star lacrosse goalie or string orchestra concertmaster/first chair is graduating, a kid who can do both will be in more demand than if the star, varsity lacrosse goalie and the concertmaster happen to be rising sophomores.
If you stand out in a subject that will be evident on your application so there is no need to reach out to teachers who in most cases do not have any direct influence on the process anyway. Musicians and athletes are valued if they are impact players. Sometimes it helps an applicant more to be very good at a sport (or instrument) that a school needs to rebuild than to be an excellent athlete (or musician) applying to a school that already has a full roster of those. It is easier to be considered to be an impact player on a weak team than on a championship one.