By now, you, Thomas Nesbitt, see this answer, you should be somewhere over 22, based on what you said and the time that has passed by. Several of the answers mention the Fields medal as an example of an award which has an age limit. So does the Nevanlinna prize. Assuming you had started to study mathematics now, add another 12 years, and because the Fields medal or the Nevanlinna prize are awarded every fourth year, the worst case scenario is that you would have only two years to come up with something extremely important in pure mathematics (http://en.wikipedia.org/wiki/Fields_medal) or information sciences (computer sciences, scientific computing and so on http://en.wikipedia.org/wiki/Nevanlinna_Prize). Certainly among pure mathematicians the Fields medal, say, is the most prestigious medal, to the extent of my knowledge. Having said that, if you care about landmarks for your career, there is no shortage of awards, both in the sense of having no age limit; the Abel prize, for example (http://www.abelprize.no/c53676/artikkel/vis.html?tid=53705&strukt_tid=53676 ) or in the sense of applied mathematics, like in the Gauss prize (http://www.mathunion.org/general/prizes/gauss/details/).
Some awards are won, like in a competition, and some others are simply given. They reach you. I won't detail any of the awards I got, because that is totally beside the point, which is that one shouldn't be worried about awards, except that when you get and if you get one, it is only a landmark of work you have been doing anyway.
As for the that period of yours where you found a "lack of intuition", I most strongly recommend the book and the short course by Keith Devlin:
"Introduction to Mathematical Thinking". The course can be found online, and you can sign in through Coursera (https://www.coursera.org/). I took it last year, and it really changed me in the way of understanding and doing mathematical proofs.
Now, my personal experience, which is not the most outstanding but hopefully useful for the issue that we got here at hand:
Probably all my life have been inclined to do mathematics, but without a formal guide. As a child I found this book of Martin Gardner "Mathematical Magic Show" (1977; in the Spanish edition it has an apple and four matches making a kind of shovel, with the idea of taking out the apple from the shovel, moving only two matches, and reconstructing the shovel). I mention this, because at the time I couldn't do long division. It took me quite some personal effort, but I wanted to learn how to do it. The algebraic concept of "unknown quantity" was at the beginning so puzzling for me, that I took the letters at face value, so I used to count their position in the alphabet, and use it for computing the final value.
At high school I was lured by genetics, so I got into a high school with medical preparation, including that subject. The school included basic calculus, and I struggled so much with a problem of limits (I think something that direct substitution would give infinity over infinity), that it triggered the first Lucid Dreaming I ever had, solving the homework, at the end. However I later learn that I could enter the Mathematical Olympiads, and prepared myself for it, with the couching of the teacher who explained all about algebra, including how to understand the "unknown quantities". I didn't get too far in the competition, but the preparation I got was so good, that when I entered the engineering school I got over all of calculus, vector calculus, statistics and so on very, very easily.
At University I made my choice of an Engineering with specialized in Biotechnology applied to Agriculture (where I did good grades in all the mathematical subjects). Still lured by genetics, molecular engineering and so on. I wanted to study, mathematically, the relationships (if there were any) of chaos, fractals, entropy, quantum mechanics and genetics, with the (extremely) ambitious idea of pondering how could a genetic problem be reversed, with the person alive and already grown up. With this idea in mind, I went to Cambridge University in the UK to ask who was working on the subject. Professor Burgess told me that no one was working directly in the mixture that I asked for, and also told me that I could be admitted to do Part III Mathematical Tripos, even coming from an Engineering degree, if I got good grades.
Once in Cambridge, I interviewed several professors of what I was concerned, and I managed to talk briefly with professor Geoffrey Grimmett. Upon hearing the list of subjects I wanted to combine, he told me "I think what you want is to find out Why do we die, isn't it?. I am not good at the big things, I work small ones [...]". But I was already in Cambridge! what to do? at the examinations of Part III, I did horrible, but still I wanted to do something. So, I found David J. Wales, and I did a C.P.G.S. dissertation "Chaos in Inert Gas Clusters" (about the Kolmogorov Sinai Entropy).
Cambridge finished, I lectured Linear Algebra for one semester, and then I managed to obtain a scholarship to do a Ph.d. in Plasma Physics (controlled fusion and/or neutron generation), at the Czech Technical University in Prague. I finished it with the dissertation "Self-Organized Structures in Z-Pinch Plasmas". A teacher over there (Jiri Gregor) told me "you should do research, and not just teaching; otherwise you will become a potato seller".
Back in my country I found no job for me on Plasma Physics, but I was offered to work for the Mexican Petroleum Institute. I found myself since then doing mathematical modeling of flow and transport in porous media, and exploring the application or applicability of concepts like fractals, quasi-hyperbolic differential equations (transport equations), inverse theory, percolation, Poisson processes, symbolic mathematics, and many others. I am not claiming that I am any closer to an expert in any of those subjects; could I be replaced by a younger, more able person? sure! but the colleagues that I have would feel a kind of "gap", if wasn't here, and those more brilliant, younger people, are working somewhere else, and in other subjects. There is so much work to do in applied or pure mathematics, that, yes, there are The Problems for outstanding mathematicians, and there are also the day-to-day nagging-odd problems that need to be solved by someone, maybe a poor sap like me.
At the moment I can tell you that I don't see myself attempting a proof Information Theory, Algebraic Geometry or one of The unsolved problems in Analytic Number Theory; and then again, who knows? I keep track on a few of them, and maybe one day I might just gather some bits and pieces, feel very, very, very inspired, and try to get my luck in one of them. Maybe a little, partial result could come out of it. Maybe never, but the point is that I have already a lot of work to do, a lot that will keep coming, some of my little, but own ideas, and if that is doing a "career" in the sense that you referred to, then there is in fact no age limit, and possibly no single path towards becoming a full-fledged mathematician; even today I wouldn't dare to call myself "mathematician", without feeling that I could be challenged by, or frown upon, by someone who did study mathematics since he or she was five years old. However some time ago, I got a colleague to whom I had a conversation about my intention to do another Ph.D., this time in mathematics, and he advised me not to waste my time, and just another paper to hang in my wall; I told him that there are people -among pure mathematicians- who do not like that I hold not a Ph.D. in mathematics neither my bizarre trajectory in studies and he answered me "I don't like your shoes". His point being that I should concentrate on the work that I did have, and not being worried about opinions of people outside our labor.
Summarizing, I tried to show you what you asked for: not just a pat on your back, telling you "everything is possible". Well, at least for me the Fields medal I am sure is impossible, because I am well over forty. But with some effort there are many other things in the course of a career that are possible, that are not unattainable, provided that you focus, relax, and keep working at it, but above all, the most important thing is that you find a subject that passions you really, so much that you will forget, at least for some moments, the difficulties of getting over an unknown subject, or a puzzling concept.