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).