Question

I am not sure how to convert my Cartesian xyz-coordinates into meters.

x = 0.666
y = 0.666
z = 0.327

a = 6,377,563.396 meters
b = 6,356,256.910 meters

Answer 1

If you are working with an ellipsoid with a=b>c, where
a=b=6,377,563.396 (towards the equator)
c=6,356,256.910 (towards the poles)
then
x=acos(u)cos(v)
y=acos(u)sin(v)
z=csin(u)
where -π/2≤ u ≤π/2
and -π≤ v ≤π

So from the known values of x,y and z, solve for u and v.
But I am not sure what you need in metres.

See:
http://en.wikipedia.org/wiki/Ellipsoid