Engineering Major applying to Pure Math Grad school
You don't say where in the world you are or where in the world you want to be. Giving this information would allow for a more targeted response, as the mathematical background of students entering a graduate program varies quite significantly by location: in particular, most European undergraduates take exclusively courses in mathematics and closely associated fields, whereas for most American undergraduates mathematics occupies less than half of all the undergraduate courses they take. I will answer with respect to an American PhD program in mathematics, since that is what I am more familiar with.
With that in mind: your overall math background seems quite strong to me. It is more than enough so that admissions committees will overlook the fact that your undergraduate major was not mathematics. That you took and excelled in five graduate courses is better preparation than the average incoming math PhD student at most American PhD programs and is similar to the average background of beginning students at, say, programs ranked 11-25 or so. [Give or take: I have been affiliated with programs ranked both above and below 11-25 so can make an educated guess, but my direct experience is more limited.]
What other math courses should I take for my application to be considered seriously?
The one noticeable omission is in analysis. Both differentiable manifolds and Riemannian geometry use some analysis topics, but far from all. Your application will be taken seriously already, but your chances would be better if you had courses like: real analysis (undergrad or graduate level), metric spaces / general topology and complex analysis. In the US system, undergraduate real analysis -- taught out of Rudin's Principles of Mathematical Analysis or a more recent equivalent -- is taken by most "serious" undergraduate math majors and is perhaps the number one course that admissions committees want to see an applicant take and excel in.
Can I take graduate courses instead to demonstrate proficiency in the courses in Question 1? For example, if I am advised to take undergraduate Topology class, does taking graduate Algebraic Topology class count as taking the undergraduate Topology class?
The answer is that it depends on a lot of factors. In the example that you give, undergraduate topology is desirable but not required. Taking graduate topology looks great...but it doesn't entirely cover undergraduate topology, depending upon what that means. In the US, unless otherwise specified, undergraduate topology is primarily general topology, which you are not going to learn much about in an graduate algebraic topology course.
Another consideration is that -- at least at the American institutions I'm familiar with -- grading in graduate courses is often different and more lax than in undergraduate courses. For instance, in more than half of the graduate courses I've ever taught, every student has received an A grade. For almost every undergraduate course I've ever taught, 25% or less of the class received an A. (It is a strange coincidence that the notable exception was...general topology, in which almost half of the small number of students taking the course got an A.) On the other hand, this is not always true. If you have recommendation letters from the faculty who taught these courses explaining convincingly why you were one of the best students in the course, you'll probably get "full credit" for your grades.
Is it bad if I took all the courses in Question 1, but didn't meet the requirements for obtaining B.Sc in Mathematics in my school? For example, my school requires that I take Statistics as part of the B.Sc requirement, but other schools may not.
Not really. Most math admissions committees care about what you know and they find that out in large part by looking at your coursework. We tend not to care so much about meeting other requirements.
On the other hand, now that I look over your story I admit to being a bit curious / confused about why your major is in engineering rather than mathematics. According to your SE profiles, you have been studying and also taking graduate level mathematics courses for the last three years. In a recent comment, you say will receive an undergraduate engineering degree in 1-2 years. It doesn't quite add up to me -- with the math courses you've taken, how could you possibly be any further than 1-2 years away from a math degree? You might want to address this point in your personal statement.
Added: I am currently the Graduate Coordinator of the mathematics department at the University of Georgia (and thus the chair of the committee that does graduate admissions).
You have a far more than adequate preparation in the area of algebra, but your lack of basic analysis and topology courses could be a problem. On the other hand, you have taken several more advanced courses in topology and geometry.
Depending upon how it is taught, an algebraic topology class may have very little to do with general point-set topology. So there might or might not be a significant gap in your knowledge. Similarly, you have taken graduate courses in differential geometry, which generally require some knowledge of analysis and topology, but it is difficult to know exactly how much you might be missing.
Some graduate programs in pure mathematics may dismiss your application out of hand, for lack of basic analysis classes. However, most admissions committees will take a more holistic view of your application. For example, algebra is your strongest area, and if your stated intention is to work in algebra, that will probably help you.
Most importantly, though, will be how your letters of recommendation read. Your will probably need at least two letters of recommendation from mathematics faculty, and these letters may be able to explain away your lack of certain courses. If the professor you had for differential geometry can write a strong letter, which specifically attests to your understanding of the important elements of analysis, then you should be in a very strong position. I would suggest that you talk specifically with your letter writers about these issues; point out to them that you are missing certain courses that would normally be taken by pure math majors, and ask whether they can highlight your knowledge of the relevant subject matter. If you can get strong letters that tell the admissions committee that you have the relevant knowledge, your chances of admission should be pretty good.
Your Question 1: Pick a couple of target schools and analyze their sequence of classes. Work your way back through the prerequisites step by step. That will help you discover any gaps in your training. These gaps might not prevent you from gaining admission, but they might slow you down once you start your new program.
Your Question 2: That would be fine, as long as the course is a good match for you. This is something for you and the instructor to decide. You can decide tentatively prior to the start of the course, and you can check in again a couple of weeks into the course.
Your Question 3: In principle it's okay if your coursework doesn't exactly match the requirements of a particular undergraduate math degree program. But you need to think carefully about where you want to end up eventually in your research and in your career, which might include teaching. If you have the time, it would be good if you could give yourself more breadth, even beyond the courses that you end up choosing by following the procedure I outlined in Question 1.