You are right strictly. The formula that should be applied is Einstein's "metric tensor".
http://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)
"The metric (tensor) captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past."
Yes, only if there is no gravitational fields.
Gravity will warp a plane into a surface in three dimensions that we cannot see or measure directly. A two dimensional warped plane cannot be fully understood in two dimensions. It is a three dimensional surface where all activity is totally constrained to the surface. A straight line in a warped space is called a "geodesic" and is the shortest "line" between two points where the line is constrained to the 3-D surface. (Our warped 4 dimensional universe can be described mathematically as a 5 dimensional sphere according to Richard C. Tolman.)
For example the shortest distance from NY to Sidney Australia is a line through the center of the earth. However if we are constrained to the 2-D surface, an airline will choose a "great circle" on the spherical earth. Girard's formula is based on triangles embedded in a warped plane which is a spherical surface. The sides (geodesics) are great circles since there can't be a straight line on the surface of a sphere.
A beam of light will appear to go in a straight line; indeed it is the definition of a physical straight line. If you set up surveying equipment on top of three mountains and measure the three angles, the sum will be greater than 180 degrees. (However in practice, imprecision and air turbulence will make this impossible to see.) You must go to a 5 dimensional description and then see that the light is actually traveling in curved geodesics because of gravitational distortion by Earth. But, again, from our perspective the lines look straight.
Velocity is a different phenomenon than gravitation, but it is another example of how unintuitive our universe is.
