If you want to do astrophysics, you will need a PhD. I recommend you get your bachelor's degree in Physics, because that itself is a strong foundation for any sub-field of physics and is more employable with just an undergraduate degree than an astronomy/astrophysics degree. Nonetheless you can still take plenty of astro courses in college but you don't necessarily need the major. Then once you graduate you can do your PhD in Astrophysics - some people even just do a PhD in physics but focus on astrophysical/astronomical research.
You need to look at the curriculum for the schools you are considering.
One of these courses was computer science, particularly dealing with programs. I was wondering if I could already start this course in high school and then not have to do it in college.
That's called an AP course, if your school offers it. Some high schools have IB programs which I do not know much about. Look at the curriculum and graduation requirements for Physics majors at the universities you are considering. Then, look to see if they offer AP Credit for certain AP Exams (usually find it by googling "University Name AP Credit"). If a good score on the AP Computer Science test score exams will fulfill a requirement for the physics major, then that is an option.
If you cannot do AP classes for whatever reason, consider dual enrollment with a local community college or university.
Computer Science is generally helpful in physics and especially astrophysics because there is a lot of modeling and programming knowledge is very versatile. Even if you can't get credit for it in college, programming is something you can learn to do early. There are plenty of resources on knowing how to code.
Nonetheless, if you are truly interested in this field, you should spend more time understanding the physics. Have you taken Physics at your school or do you plan on taking it? Don't worry about trying to knock classes out before college too much - if you work hard you will be able to graduate in 4 years.
Also, is there any other courses that may be required to become an Astrophysicist? (Particularly engineering)
as I said before, you need to look at the curriculum of the physics programs at universities you are considering attending. They will explicitly tell you what you need in order to graduate with the degree. What do you mean by engineering? I don't know of any physics/astronomy program that explicitly requires engineering courses. Physics majors can however choose to pursue engineering in grad school, but there is quite a bit of remedial work to transition into engineering.