Does a mathematician study more, or research more?

It is very important to convince oneself to view the good work others have already done as potentially helping you, not being a burden.

It is observable that the "school-work" model of mathematics mostly presents us with obligations-to-study which are not well explained, apart from the usual threat-of-bad-grade and/or loss of funding. And, indeed, some of the traditional requirements are rather stylized, and have drifted over time, or have fallen out of sync somewhat with contemporary events, so the underlying utility can be obscured. But one should not be deceived by this picture of mathematics presented by "requirements" and such. Some things are very useful "even if they are required". :)

As some consolation, also the very model of "study" presented by school-math is itself considerably caricatured, in my opinion. The idea that one is not allowed to move forward without having done all the exercises and assimilated all the proofs of all the lemmas is needlessly and unhelpfully constrictive. Certainly not helpful in getting any larger perspective. Many of the usual exercises are merely makework, artificial, and not a good investment of time. A more mature and useful notion of "study" is to try to acquire awareness of the general pattern of events, some illuminating examples, and only return to low-level or foundational details when they become "action items", sort of thinking in terms of need-to-know.

That is, imagine there's no final exam, no quizzes, no weekly exercises to be graded, but that one should try be able to answer the "What's the point of this?" questions.

At a further point, if one wants to make genuinely useful contributions, genuinely advancing collective understanding, it is obviously necessary to have some awareness of what that collective understanding is already. Re-inventing things can be fun, and is inevitable, but one wants to do more.

In fact, I would argue that (a mature notion of) "study" is inseparable from (a mature notion of) "research". Or indistinguishable. In the endeavor of trying to improve one's understanding of some phenomena or structures, by looking at what other people have done and trying to organize it in one's mind, often one "accidentally" understands something that perhaps was not already well understood. Bingo: "research".

Can someone tell me if professional mathematician really have to spend so much time into reading what someone else did?

What a very strange way of putting it! If you care about the truth in some domain of enquiry, why wouldn't you want to spend a lot of time exploring what is already known, what has already been discovered by the brightest and best? Why wouldn't you want to equip yourself with as much relevant knowledge and understanding as possible? What better way of putting yourself in a position to have the widest range of the best tools available for extending knowledge in the area?

I can't answer your question since I'm not a professional mathematician, but your question sounds like a cry for help.

A lot of young people are dreaming of making their passions to their profession but have to suppress their real interests to become competable. And that's hazardous: you risk to kill your passions to become a cog in an efficient machinery dealing with tip knowledge that interests somebody else but not really you.

Paul Garrrett wrote that study and research was inseparable and I believe that having fun while studying should be inseparable too. There must be time for playing around. Keeping the passion alive make the study much more giving. The right balance between passion and work, I think, need to minimize the destructive impact from careerism and social pressure.

After all, you have a talent to manage and to invest in, in a sensible way.