* Is there any concept of “parallel lines” in spacetime geometry that has any meaning or definition?
I think you have two possible generalizations of "parallel lines" from Euclidean geometry.
1) Initially parallel geodesics. Since geodesics are the straight lines of curved manifolds, they have constant
tangent vector when computed with a geometrically-informed version of the derivative. Using the generalization of irrotational sliding called
parallel transport, we may take a path to a point off the first geodesic and establish a second geodesic with the "same" (parallel-transported) tangent vector. At least we can do this in well-defined manner in the limit of short paths, for the result of parallel transport is path-dependent when the manifold is curved.
Example on a globe. The prime meridian and the line of 1° W longitude both run purely north-to-south at the Equator (and everywhere else), which because they have the "same" tangent vector (moving purely north at the rate of one meter per meter of arc length). You can parallel transport such a vector along the geodesic of the equator. But if you parallel transport the prime meridian 179° along the Equator, it will now be anti-parallel to the geodesic which is both 1° W longitude and 179° E longitude. Likewise if you parallel transport the tangent vector of the prime meridian via a geodesic which is not the Equator, the new geodesic will not be a line of longitude except in the limit of a short separation.
Example on a globe. If you don't use short geodesics, parallel transport can turn a "north" tangent vector of a line of longitude into the "east" vector of the Equator. (Hint: Use a
big circle centered on Quito, Ecuador.)
2) Curves of fixed geodesic separation. Let A be a curve, and L be a length, and n be an arbirary tangent vector. Then along every point of A we have n' which is the parallel-transported tangent vector. Thus at every point of A we can start a new geodesic of length L and those endpoints form a curve B. If A is a geodesic, then B is not expected to be a geodesic.
Example on a globe. Let A be a circle of radius 500 km about New York City. Let n be "north" as defined at the south-east extremal point of the circle and L be 50 km. But n' is not "north" anywhere else. Then B is
a smaller circle than A a smooth curve which superficially resembles a circle centered 50 km north of New York City.
Example on a globe. Let A be the Equator (a geodesic). Let n be "north" as defined in Quito. Then n' is always "north". Let L be 5000 km, then B is the line of 45° latitude, but is not a geodesic.
Only in flat geometry can you have Initially parallel geodesics that have fixed separation, thus never cross or diverge. Since all space-times are approximately flat in the limit of small regions, it follows that on a globe or in space-time we can work in the approximation of flat geometry. Thus most square buildings don't have corners that can be measured to be more than 90° and terrestrial gravity never enters into discussion of atomic phenomena lasting only tiny fractions of a second.
** What other candidate properties are there that might determine or be affected by such a putative force?
The sky's the limit. We have evidence of fundamental particles have such intrinsic properties as mass, electric charge, angular momentum, electroweak charge and chromodynamics charge. Recently, evidence that fundamental mass is really a measure of a type of charge which is coupling with the Higgs field was shown. But non-fundamental particles, like protons, have a mass which is not explained by the Higgs mechanism, but rather in the coupling of quarks together to form a colorless composite particle. Even though the types of mass are theoretically distinct, their behavior under gravity is identical which suggests that behavior in a gravity field is not a force but something more fundamental like the behavior of geometry.
*** Not quite sure what that phrase means.What are masses "composed of"?
Atoms. Each atom is composed of electrons, protons and neutrons, but the mass of each atom is less than the mass of free constituents and varies significantly between isotopes. But don't appear to care about composition to the limits of testing.
As a generality might it be correct to say that any model of a particular n-dimensional reality requires a n+1 dimensional "standpoint" in order for the human mind to visualize it (but not to mathematically describe it) ?
Space-time curvature does not imply that there is some higher dimensional reality that the space-time manifold is embedded in. While the math of curvature of manifolds was developed assuming the manifolds were embedded in some Euclidean space, nothing in the math requires or depends on such a superspace. Also, 5 dimensions do not suffice to embed all possible space-time manifolds in GR, so even if your meta-physics were correct in concept, it is wrong quantitatively.
// Edit, inserted "big" in hint for second example. Fixed third example
Your desperation tactics continue: The onus is on you to show that the incumbent model of GR, curved spacetime and all that goes with it is invalid.
