There's no such thing (at least at this level) as being "good at math" or "not good at math." There is only "learns math faster" and "learns math slower," "knows more math" and "knows less math." That's it.
If you're someone who learns math slowly, then, well, you'll need to spend more time studying. As to whether there's enough time in the semester to achieve the competency you desire, or whether you want to spend that amount of time, that depends on your goals and your expectations.
I don't know whether or not you're "cut out" for the university math degree -- in part because I don't know how rigorous the math degree program is at your university, and in part because (again) it depends on your standards and your willingness to put in the time/effort.
As to how to study calculus better...
Well, if you're struggling with "90% or more of the questions on homework," then you need to ask yourself whether it's the actual calculus you're struggling with, or whether your algebra / pre-calculus foundation is shaky.
If it's that your algebra / pre-calculus foundation is shaky, then I'm afraid you're in a tough situation. The only thing I can suggest is to get a good tutor who will patiently work through the basics with you, from scratch. Above all, you need to develop not just competence, but fluency with the mechanics of algebra and the techniques of graphing functions.
If your algebra and pre-calculus is solid, but it's the calculus itself that you're having trouble with, then things won't be quite as bad. If it were me in that situation, I would start from the beginning of the semester's notes, from Section 1.1, and for each concept ask myself "do I understand X"? If not, what exactly don't I understand about X? Then I would do lots of problems from that section. If I got stuck, I would get help.
Then I would do the same thing for Section 1.2. And so on and so forth.
Then after all of that -- understanding concepts and doing textbook / homework problems -- I would do many problems from previous exams (if available). If previous exam problems are not available, then you should try to do hard problems from the text.
Ultimately, in my (admittedly limited) experience, students' issues with learning calculus often come down to one (or more) of the following:
- Shaky foundation in algebra / pre-calculus
- Not memorizing the formulas and rules
- Memorizing the rules but not practicing them enough
- Settling for a vague, hazy understanding, rather than developing a clear and precise understanding
The last point applies in particular to theorems like the Intermediate Value Theorem. Many students have a hazy intuitive sense of what the theorem says, yet few can actually state the theorem correctly. As such, it's no surprise that they often misapply it.
In summary: success in calculus (and in most math classes, for that matter) require both (1) technical competence in calculations (or proof techniques), and (2) an intuitive yet clear understanding of the concepts.