Would that concept of sphere and flat change if we discovered and moved into 4th, 5th, 6th or more dimensions?
no. those other dimensions are simply extensions of the previous ones.
1st dimension is the number line, aka the x-axis
2nd dimension is the cartesian plane which is nothing more than two number lines. one going left and right (the x-axis) and another going up and down (the y-axis)
3rd dimension is a cartesian plane which is nothing more than three number lines. one going left and right (the x-axis), another another going up and down (the y-axis), and one going from front to back (the z-axis) or into and out of the screen.
in each of these dimensions the plot points or the collections build on each other, but remain what they are in their own dimension.
1st - you either have a point, a collection of scattered points, or a collection of connected points(the line)
2nd - the integral of the point from the first dimension is a line in the second and that line in the first becomes and area (squre, circle) in the second. this does not take away the fact that when you take the derivative and go back to the 1st dimension, you are once again left with a point or a line.
3rd - the area becomes a volume (cube, sphere, etc)