Before we go much further with the hard stuff, let's see if there is any way to know whether we are constrained to such a curved or "warped" [hyper]surface.
For the "bug on the balloon" there certainly is: simply check whether
Euclidean geometry (trigonometry, ) works properly on figures in the "plane"
of the Earth's surface. As an extreme example, note that two "straight
lines" which cross at one point on the Earth will cross again
on the other side! Also note that one can make a "triangle" out of
great circles in which all three angles are 90o!
[Just make the length of each side equal to 
the circumference of the Earth.] And so on.