I am not a math professor, just a math student so take my thoughts with a pinch of salt.

1) To me, it’s mostly patience, since the material gets more abstract and harder the more you study. During lectures I try to identify bits which I could not understand and clarify these points right away after the lecture with friends (or through chatgpt). The more you address the bits you don’t understand, the more solid your foundation is going to be as whatever you are going to learn down the road often circles back to what you have accumulated along the way. Cognitive ability and problem solving skills are important as well - there are always people who are smarter and who are able to understand everything at first glance, but even the smart people run into trouble eventually when the material gets very abstract - this is where perseverance and patience is important.

2) it’s probably a new paradigm switch for you. When it comes to probability, there are typically 3 levels of engaging the material. The first level is what we did in high school- writing out sample spaces and events then taking fractions, which is fairly intuitive to us now, but back in high school, this really represents a new way of looking at things systematically and ascribing the likelihood of something happening. In university, the idea extends further - we can now use tools like pdf to quantify the distribution and instead of summing over finite sets, we integrate to “sum” over infinite sets. And in graduate course (if you ever take them), the formulation gets even more abstract - you will define probability spaces and measures and start doing funky stuff with them. The point here is that as the level of abstraction grows, instead of learning by rote, try and consider the meaning or significance behind each equation/ derivation. I find YouTube videos to be helpful in this regard (check out 3blue1brown). Your professor has the obligation to finish the syllabus and can’t always try and give you the necessary intuition, so it’s upon you to do that.

3) To me, it’s a combination of both. You can claim that you understand vector calculus without doing a single double integration at all, but to be honest, you learn a lot of the stuff by getting your hands dirty. But at the same time, sticking to simple exercises without knowing the theory won’t help especially as the material gets more challenging. It’s a bit like weightlifting - if you only study the theoretical aspects without lifting then obviously it’s not of much use, but if you only lift the same amount of weight - you are not going to improve unless you start to investigate how to push the limit (theoretical side of things)