Am I fit for higher studies/teaching in mathematics? Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I should focus on earning as soon as possible), and so instead I opted for an engineering degree. While in college, I used to find myself fascinated by higher math and other stuff, and used to score near-perfect all the time (but only in mathematics!). I must make it clear that I don't have an extraordinary talent for math; it's just that I enjoy exploring it very much, and then describing it to others in interesting ways. Anyway, it so happened that I went on changing job after job and was never satisfied. Now at the age of 27, I'm sitting home and realizing that I should have given mathematics a thought. So I'm thinking of taking up a graduate course and picking up where I left off.
HOWEVER . . .
I dream of becoming a teacher cum researcher some day, even if it's at school-level only. Before my graduate course begins (in 2-3 months), I've started reviewing math from my school books. Now the point is that I expect myself to be much more mature and smarter by now. That means I should be able to solve any problem and prove any theorem given in school math. But I can't, and it's shattering me. I mean, if I can't even prove basic theorems related to Euclid's geometry, how can I ever hope to do some authentic research like the mathematicians I so admire? This makes me wonder if I’ll ever be fit for teaching. If, at the age of 27, I can’t even master basic school mathematics with certainty, how can I ever hope to tackle problems in calculus and polynomials that students bring me tomorrow? It’s as if I’m cheating myself and those who’ll come to me for instruction.
Am I being too hard on myself? Am I expecting too much too soon? Does there come a point in a person’s math studies when he is able to discern properties and theorems all on his own? Or are all students of mathematics struggling and hiding their weaknesses? Or I really have no talent for math and it’s just an idle indulgence of mine. I mean, I’m not sure “how good” I’m supposed to be in order to feel confident that I can pull it off. In general, I find myself wondering how much do the others know math. Do all the teachers have perfect knowledge of, say, geometry, and can tackle every problem? If not, what gives them the right to call themselves teachers?
I have a feeling most of these questions are absurd, but I’ll be very thankful if someone can put me out of my misery.
 A: 
"I mean, if I can't even prove basic theorems related to Euclid's geometry, how can I ever hope to do some authentic research like the mathematicians I so admire? "

These are different skills.   There are many researchers who (as students) could solve all the problems in the calculus or geometry books, and now cannot as easily find the answer when students come asking questions. Hence the reliance on graduate student assistants, solution books, and permanent calculus lecturers. 

"This makes me wonder if I’ll ever be fit for teaching. If, at the age of 27, I can’t even master basic school mathematics with certainty, how can I ever hope to tackle problems in calculus and polynomials that students bring me tomorrow"?

An overestimate of the requirements for being a school instructor!  Only a small fraction of teachers can explain the mathematics credibly + know the theory behind it + can solve hard problems, and fewer have all these skills before working as a teacher.  Some of the most effective teachers begin with relatively little advanced knowledge, but love the subject and constantly deepen their understanding by reading, solving problems, using math software, taking classes, assisting with student research projects and competitions ... or participating in online math sites.   
A: I took a bachelors in software engineering at an engineering college. 3 years coursework/projects and a 6 month obligatory internship -- I'm from Denmark in Scandinavia so YMMV.
We had calculus, algorithms, discrete maths and circuit analysis. We had theory and application but with little emphasis on the proofs come the exams. 
When I graduated I continued to pursue a Masters degree at the local university's faculty of computer science.
Now everything is about proofs and I have had more maths than I thought I could bear.
I REALLY struggled with the proofs at first, but to me it comes down to method and practice. You have to get comfortable with various ways of proofs: by induction, by construction, by exhaustion etc. I think it's because much of the basic prerequisites of being a mathematician isn't clear to "outsiders". It certainly wasn't, and still isn't, for me.
I find that I cannot follow advanced courses on certain subjects, if I haven't done introductory coursework that was related. Most of that comes naturally by the course having prerequisite courses, but it still is a factor.
Personally I do great in courses that interest me, so if you're hungry for learning maths, I say go for it! I think you'll do good and have fun at it. Don't let it scare you away - math is very hard to learn for many people, and understandably. Many branches are very abstract and hard to visualize mentally. Just be devoted and take your time, and don't be afraid to flunk a course or two. 
I think that if you've never flunked or just baaarely passed a course, then you've never challenged yourself hard enough.
A: To be honest I was never good at high school geometry and geometry in general. I am also interested in mathematics and want to pursue a PhD in pure math. I always thought there was something wrong with geometric arguments -- and it turns out there is. Many philosophers and mathematicians explained why they are not sound using examples. You can use geometry to prove false statements. I remember when I took real analysis, I understood most of it, but never felt confident about it. But know it all makes sense. The same goes for complex analysis. It all makes sense when you use it in, say, number theory. Trust me, no mathematician, however great, could prove every known theorem in her field without using a book or two. Of course, after proving a few basic results you will be able to distinguish between one line proofs and one paragraph proofs. If you feel like you cannot prove something, just go back to the definitions and familiar results. Write everything in full i.e. the definitions, theorems, etc. and after a while you will skip those steps. I know some people who claim to love math, but they are also majoring in something like english or chemistry. If you are truly passionate about math, you should not major in anything else, and you should never do math for a potential employer. If you finally realized that math in the only thing that you are interested in, then you should just go for it.
A: I don't have an answer, but I have a suggestion for finding one.
Most mathematics professors will allow you to visit lectures and exercise classes, and even welcome you if you show more enthusiasm than the average student (trust me: not hard to do). You won't get a degree that way, but you will get a better idea if studying mathematics is something for you.
A: As a grad student, I think that you should stick to engineering.  The jobs are just so more plentiful.
