Ah. Thank you.
I find it amusing that you can (correctly and meaningfully) say “there really aren’t that many” about something of which there are infinitely many (one for each integer n >=0 and each k >= 1). 
The number infinite, but it isn’t one of the big infinities.
Isn’t a Klein bottle of the same genus as a torus?
Technically speaking, there’s no such thing as the surface of a sphere, or at the least it’s redundant, because a sphere is defined as the set of points at some distance from a point and excludes the interior points.
A ball however includes the interior points. The surface of a 3D ball is a sphere (which is a 2D surface).
Obviously these are the mathematical definitions and they don’t quite match the common English definition. Just something to watch out for.
I think not. Torus is (orientable) genus 1, Klein bottle is (non-orientable) genus 2. A cross-cap is non-orientable genus 1.
Let me point out that the interior of a flat disk (all the points not on the boundary) also constitutes a finite 2 dimensional surface without a boundary. Oh, you meant a compact surface? Then Leahcim is correct, of course. No, I am not going to attempt to define compact.
Okay I lied. Although this is not the general definition, a compact surface is characterized by the fact that every infinite sequence of points contains a convergent subsequence. On the interior of the unit disk, the points whose distance to the origin is less than 1, the sequence of points (0,0), (1/2,0), (2/3,0), (3/4,0), … does not converge since its limit, (1,0) is not in the interior of the disk.
So, an ordinary sphere is two-dimensional. There is such a thing as a “three-dimensional sphere”, but it is NOT the three-dimensional “ball.” A good overview of such things for the layman is The Poincaré Conjecture by Donal O’Shea.
Nitpick: While local regions of 2-D manifolds have two coordinates, I don’t think good global coordinate systems exist in general. The torus may have a better such system (latitude and longitude both circular 0→2pi) than the sphere has, but can anything nice be conjured up for a two-holed donut?
Every closed surface can be described by taking a polygon and identifying edges. For example, a two-holed donut can be gotten from an octagon. So, put a nice set of coordinates on the octagon and that gives you global coordinates on the two-holed donut. Of course, there will be singularities where you identified edges, but that’s true for latitude and longitude at the poles of the sphere, too.
In a simpler common sense approach, the surface of a sphere would be measured in square units like square inches. That makes it 2 dimensional. In the same way, its volume would be cubic units like cubic inches therefore 3-D, but a circumference would be just inches so it is 1 dimensional.
You are perhaps thinking of the fact that they both have the same Euler characteristic (specifically, zero). But genus is different from Euler characteristic. Genus is sort of an odd concept, but basically, the genus of an orientable surface is how many toruses are joined together to make it (every orientable surface being decomposable in this manner, and toruses not being any further decomposable), and the genus of a non-orientable surface is how many projective planes are joined together to make it (every non-orientable surface being decomposable in this manner, and projective planes not being any further decomposable).
And by “joined together”, I mean via the “connected sum”…