A Former Atheist

[Alex K]

...the solar system would have to travel awful fast to make the trip.  [T]hat would have the sun traveling at a little over 3% of the speed of light.  Should the sun travel around us, we'd fall in.

This gets into the complicated issue of reference frames. In physics all motion is relative, so it doesn't matter whether you think of a rotating earth in a stationary reference frame or a stationary earth in a rotating reference frame. In either of these frames the relative speed of sun and earth is simply the 30 km/sec we expect from the earth's orbital situation. The only thing that's going 3% of the speed of light is the imaginary axes of the rotating frame at that distance. If you extend them out far enough, they appear to go faster than light in violation of Einstein's Special Relativity.

James Callaghan explains in The Geometry of Spacetime: An Introduction to Special & General Relativity (Springer, 2000,  §4.1, pp. 152-153): There are equations that convert coordinates between the stationary and rotating frames. Rulers measuring length in the transverse direction are foreshortened by a Lorentz transformation when converting. This eliminates the huge apparent speeds.

As far as the orbital motion goes, again it doesn't matter which body moves around which: it's relative motion that counts, and  earth is in free fall in either case. The orbit is a conic section, for earth a nearly circular ellipse. As long as we have that 30 km/sec transverse component, we don't fall straight into the sun.

Feel free to correct me on any of these matters if I'm wrong; I'm not expert at this stuff.  

Yes, what you say sounds all correct (although I didn't check the numbers), and the question what it means to be stationary is really subtle in General Relativity. One can indeed easily go to a spacetime-dependent reference frame in which the earth is fixed at a certain coordinate position while the sun rotates around it. This is allowed in GR because it is set up to be completely coordinate-choice independent by design.
If you make this coordinate change, the transformation of the metric tensor into the geostationary rotating reference frame causes the introduction of large nonvanishing Christoffel connections which introduce "gravitational forces" pulling the entire universe towards the earth such that it rotates around earth once every 24 hours. This metric grows without bounds as one goes further and further away from the solar system. At the end of the day, the coordinate system which is special from a purely mathematical perspective is the one where the metric does not diverge as one goes away from earth, and this is more or less the same as a Newtonian inertial frame where the Galaxy is unaccelerated and slowly rotates.

One can do the analogous thing in Newtonian physics and choose a rotating reference frame. The only difference to GR is that one has to live with a change of the mathematical laws, in the form of new coriolis and centrifugal forces. In this case, we have a unique notion which systems are special, namely those in which the newtonian laws have the simplest form and all pseudo forces vanish. The eerie aspect of GR is, as I said, that it is not immediately obvious from the equations which frame is "correct". Einstein fought a lot with the problem of coordinate invariance exactly 100 years ago, and he was afraid that his theory didn't make any predictions because of the huge freedom (His "hole problem"). Fortunately, he figured out how to extract physical information despite the coordinate freedom.

I think we should all agree that I am not a cutting edge lactating bull sitter...

Lacerating? Or milk?  

You underestimate my potential for bad and/or nonsensical jokes.