I have just turned 18 and am in calculus BC (calc 1 & calc 2). I most certainly grasp and understand the concepts of calc 1 however every once in a while a I seem to struggle with the calc 2 work. More often than not I stay after school and end up understanding the concepts and/or my mistakes and why I made them. I'm a very anxious person so I if I don't understand something I worry that I cannot become a professor (this is a huge goal I would love to achieve). Is there any advice for self confidence and any advice at my chances of working at an University later in my life. Thank you!
A lot of students seem to make it through high school and well into college with the idea that school is supposed to be easy, and that having to work hard, or being confused at times, or struggling with some topics means that something has gone wrong.
This is nonsense. Struggling with material is a normal part of learning. Indeed, it's an essential part of learning, because everyone eventually hits a point where they struggle, and learning how to deal with is an important skill.
Relatedly, you almost certainly don't understand the concepts of calc 1. That's not a criticism; it's basically impossible to fully understand the concepts of calculus 1 when you've just taken it. Learning math is a cyclical process, where you learn concepts, use them as a basis to learn additional concepts, and then return to the old concepts and learn more about them in light of what you now know.
I bring this up because one thing you should expect while you learn calc 2 is to have to revisit things you think you were done with and realize new things about them.
To answer your second question, we have no information about your chances of working at a university later. No information. Neither the fact that you breezed through cal 1c, nor the fact that you're struggling a bit more in calc 2, is a particularly strong indicator, especially since calc BC is very unlike the sort of math professors do.
Henry is completely right that what your struggle is normal. I want to add that your reaction to it extremely positive. Keep doing what you are doing! Knowledge cannot be directly transmitted from your teacher's head to yours; it must be translated into a communication medium such as words or pictures. There is always something lost in this process. That is why people make mistakes or form misconceptions.
By staying after and carefully considering every mistake and misconception, you are rebuilding what was lost when the knowledge was first communicated. Consistently doing this both in and out of class is the key to truly learning from the course.
You started this practice motivated by your desire to stay competitive for an academic position in the future. Stay committed to your this approach to learning. If you continue building the habit of consistently filling all the holes you find in your knowledge, you will develop confidence that you thoroughly understand all of the material presented in class. As you continue doing this, I think you will find that desire to be confident in the material will overtake anxiety as the motive force for your studying.
I'd also like to add that if you intend to be a teacher, your struggles now will most likely make you a better teacher because you will have a better sense of what your students go through and you'll be able to share your coping strategies with them. In fact, if being a good teacher is one of your goals, I suggest you watch how your fellow students cope as well for further ideas to help your future pupils.