Wrt fitting in the subjects - high school is a four year period. There is no exam midway through after which you narrow down your subjects.
A high school day in my local HS goes from 8-3:15. Students do a homeroom period of 10ish minutes, followed by eight periods of class. Three of those periods are taken up by lunch, PE, and study hall. That leaves five academic periods. Some students who do music performance miss study hall and do music instead. You do five subjects every day therefore.
An example of a course load for one year (two semesters) would be:
English, maths, history, biology, Latin.
The following year you would do chemistry instead of biology and the year after that physics and psychology instead of chemistry and history. For your final year you might drop Latin and do studio art, US government, environmental science, maths and English.
There are endless permutation and combinations of courses possible, with over a hundred courses to choose from, and on top of that you can do summer school to fit in art courses or to advance a year in maths, or do a history course. You can also repeat a core subject course if you failed it during the school year.
Kids who are going to university have to have a minimum of three years of maths and English, a mfl or Latin, at least two years of history/ humanities, and at least three years of lab science. Kids who want to get into a good university will do their courses at honors or AP level.
Students don't progress through school in a cohort of their peers - classes can be composed of students of any age from 14 to 18. Students are placed in classes according to aptitude and/or previous grades, not age. The exception is drivers Ed, which is for students who will turn 16 by the end of the school year.