Somebody clever once said you don't get to be a concert pianist by watching somebody else play the piano. I would say that the exercises are crucial because it is too easy to read over someone else's writing - particularly that of a good teacher and think that you have it. There is however a diminishing return on just doing exercises based on a single topic at the end of a chapter on that topic. Once you have it in your mind that the answer is in the chapter you have just read it acts as a big clue and you arguably stop making the connections between different areas. I would do between 5 and 10 exercises on each chapter to clear up any misunderstanding and then move on.
You really don't have to be a genius to be good (really outstanding might be different) at maths. I earned a 1st class degree from a decent university and always considered myself to be a bit thick compared to other people on the course. That was based on 50 - 60 hours per week, every week for 3 years. A little less during holidays and considerably more coming close to exams. That was enough to learn a reasonable enough range of mathematics to solve a range of problems. That probably broke down as 10-12 hours lectures another 10-12 hours discussing lectures and the rest problem solving.
The question of what is hard work in maths is complicated a little by the fact that most people who learn it to the level you are talking about love the subject. The work may be hard but the work isn't actually work. People naturally fall into spending that amount of time on it because that is their leisure time as well as their work time. If you have the passion you will find yourself putting more and more time into it.
The question of other fields and tools is really one of specialization. It is talking about within maths or maths related disciplines and that kind of specialization is really graduate level/ doctoral level study. An undergraduate course will provide you with a number of different courses and broadly speaking set out the amount of time you spend on each. Even if you don't have the resources or the time to do that kind of course, download a syllabus for a first year course and see what is required. There might be between 6 and 8 topics studied in a given year.
Finally I would say - depending on your experience you may find that mathematics at university level is not quite as numerate a subject as you might expect. The sort of calculus that I think is in Stewart certainly has its place on a maths course but you could find it in an engineering or natural science course as well. Certainly "Pure" Mathematics at higher levels tends to work in the abstract and takes calculus as the foundation for abstract spaces of objects. Applied mathematics tends to be a little more like the mathematics in Stewart's book.
This exchange here is all the opportunity you need to talk to other mathematicians.