It depends on the course of studies laid out in your particular degree program. I had a major and a double minor in music, and I had to take no math classes at all in college. But in a Liberal Arts curriculum, I'm almost certain you'd have at least one math class.
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The math class(es) you are required to take in college are based on your area of study, so someone studying engineering may have to take certain math classes that a chemistry major may not have to take.
Usually you'll take it in 12th grade and college, but some highschools will let you take it earlier
No. College Algebra and Precalculus (or Trigonometry) are enough preparation to take Calculus.
College Algebra is the required math for all college students.
Calculus is higher than Algebra. There are also courses on Linear Algebra and Differential Equations that are higher than college Algebra. If you contact the Math department of any college they should be able to give you a specific answer as to what courses they correspond with and what a challenging math class would be.