You raise many interesting points, Ken! Some quick feedback:

I agree that is a common pitfall, one you and I try not to fall into!

Not sure I understand (even after reading what follows), but FWIW one point I have often tried to emphasize (following Bonnor and others) is finding solutions (typically one determines local solutions and tries to find maximal extensions) is only the beginning; the hard part is often finding a plausible physical interpretation. Einstein himself had quite a bit to say about this (see for example "hole problem" in D'Inverno's textbook) but unfortunately the proper mathematical tools were not widely available in his time.

If you want to learn more about exact solutions of the EFE, I recommend that as your first stop you look for a two part paper by Bonnor (the second part has coauthors), which appeared years ago in General Relativity and Gravitation (I can provide the exact citations if neccessary), in which he emphasizes the importance of finding a physically plausible interpretation. A good on-line survey of exact solutions which also stresses this point is http://arxiv.org/abs/gr-qc/0004016 (two more recent papers by Bicak are much less extensive). The standard source for exact solutions is the monograph by Kramer et al, Exact Solutions of Einstein's Field Equations, which is a veritable gold mine of information but which is unfortunately hard to read, particularly concerning some of the most important solutions. A prerequisite for reading this would be some graduate level gtr textbooks plus the monograph by Hawking and Ellis, The Large Scale Structure of Spacetime. An excellent specialized monograph is Griffiths, Colliding Plane Waves, which the author has made available on-line at http://www-staff.lboro.ac.uk/~majbg/jbg/book.html
(There is a huge huge literature so these are just a few of hundreds of particularly valuable resources.)

Just so.

I was actually referring to a number of distinct lines of thought which I lack time to even try to truly explain. One idea which should rock your universe is a (not entirely rigorous) argument by Ted Jacobson that the EFE can be derived by demanding that dS = dQ/T hold for local "Rindler horizons"; see
http://arxiv.org/abs/gr-qc/9504004
http://arxiv.org/abs/gr-qc/0602001
Another is the suggestion that the gtr can be viewed as a general framework for studying how energy gravitates, in the same sense that thermodynamics can be viewed as a general framework for studying how energy can (cannot) be used to do useful work. Both emerge, it is argued, from fundamental "theories of matter" (and/or fields/interactions), but conversely, both rule out putative fundamental "theories of matter" which fail to be "thermodynamically self-consistent".

You should be interested in numerous long-ago postings to venues like sci.physics.research in which I tried to explain similar insights. I'm sure I've put this much better in the past, but my points included:

1. Physics seems to demand choosing a manifold category (C^p spacetimes? C^infty spacetimes? real analytic spacetimes?) for convenience of the moment, which raises troubling issues related to "Will the real gtr please stand up?" See for example arXiv:gr-qc/9507019 and the monograph by Griffiths for some related discussion.

2. The EFE places no limits on what can stand on the RHS. Einstein himself understood this pragmatically as "any energy-momentum tensor which is physically reasonable according to well-established physical theory", an attitude which would be hard to maintain today with so many wild and wooly proposal being touted in the absence of convincing experimental tests. It is widely appreciated that the various "energy conditions" (see for example Carroll's textbook) are inadequate attempts to propose a crude "filter", since there seems to be at least one experimentally well-established physical scenario which violates each of these conditions (consider for example Casimir force). This "vagueness" perhaps be can be understood in terms of Jacobson's ideas.

3. By its very nature, the EFE makes it hard to set up allowed initial conditions, particularly if you are looking for model in which two things actually interact in a physically interesting way. (For example, currently an exact solution modeling a binary black hole system seems entirely out of reach, although in principle such things clearly must exist in a suitable space of vacuum solutions.)

4. As everyone here probably is well aware, much effort has gone into developing formalisms for treating splitting the EFE into constraint equations (criteria for "legal initial conditions", including geometry of an "initial hyperslice" and initial values of physical fields on that initial hyperslice) plus evolution criteria (evolve the geometry of a family of hyperslices plus values of physical fields on those hyperslices). In particular, theorists seek formalisms in which the evolution equations have desirable properties in terms of the theory of nonlinear PDEs. This program must confront numerous obstacles: one of the most fundamental is the development of Cauchy horizons. Particularly striking is the fact that it is possible to write down simple exact solutions in which such horizons are also "weak null nonscalar curvature singularities"; Ellis and Schmidt have given (not entirely rigorous) arguments that small objects might survive an encounter with such singularities, but by definition gtr proclaims its failure to make unique predictions about what might happen after that!

That's pretty much what I thought you meant. I usually put it something like this: "in the weak-field slow motion limit [mathematical details ommitted] we recover the non-relativistic classical field theory formulation of Newtonian gravitation from gtr". Similarly for classical limits of QFTs.

I avoid saying that classical mechanics has been -replaced- by quantum mechanics, since it is -appropriate- to use the former in many circumstances not merely for computation convenience but for (valid) theoretical insight. Old bad theories (like N-rays) are well and truly dead, but old good theories never die, they simply are further developed as an important "sector" of science, even if they are in some sense "subsumed" by "more general" theories.

I keep changing the subject, but interestingly enough, a very interesting arena in which one can study in great detail how properties emerge is in Kleinian geometry, including finite Kleinian geometries. As a very simple example, consider how the projective plane over GF(q) relates to the affine plane over GF(q). It is best to start by considering these as point sets equipped with actions by certain groups, the symmetry groups of the geometries. The symmetry group of the former is PGL(3,q) and the symmetry group of the latter is the subgroup AGL(2,q), namely the subgroup which leaves invariant some choice of one line in the projective plane as "the ideal line" (aka "circle at infinity"). As a general rule, when you restrict to the action by some subgroup, some of the orbits under the original action will split up, showing that the "more rigid" geometry (affine geometry, in this comparision), makes finer distinctions and has more invariants (all other things being equal). For example in the case GF(3), the projective plane has 13 points and 13 lines, but restricting from PGL(3,3) to AGL(2,3), the orbit of points under PGL(3,3) splits into two orbits under AGL(2,3) (9 ordinary points plus 4 ideal points), and the orbit of lines under PGL(3,3) also splits into two orbits under AGL(2,3) (12 ordinary lines plus 1 ideal line).

But that's only the tip of the iceberg: according to Galois we should study the stabilizer-fixset lattices, and then (particularly when we add actions on lines to the mix) we can see that elements of these lattice enumerate the "geometric figures" which are possible in each geometry and how they are related. For example, we can see from these lattices that in the projective plane over GF(3), each point lies in 4 lines and each line contains 4 points, while in the affine plane, each ordinary line contains 3 ordinary points and 1 ideal point (the "point at infinity"), while each ordinary point lies in 4 ordinary lines. Following Boltzmann, Klein, and Planck, certain homogeneous spaces (complexions) then represent the possible "motions" (in sense of projective or affine geometry) of each kind of geometric figure (for example, the four lines concurrent on a point, in case of a projective plane), and numerical invariants of these are entropies satisfying the axioms laid down by Shannon. For example, in projective geometry, if we fix two points we still have some freedom to move the other points on the line through these two points, but in affine geometry, the entire line is pointwise fixed; these distinctions are reflected in the fact that the complexion of two points in affine geometry is smaller the complexion of two points in projective geometry (which again illustrates the notion that affine geometry is "more rigid than" projective geometry).

Even more striking observations occur when we ask how much information we gain about the motion of a line when we learn the motion of a disjoint point (for example). We can say that "Shannonian" entropies (conditional entropies), numerical quantities which obey certain axioms formulated independently of any probabilistic context, arise as numerical invariants of underlying algebraic objects, the complexions (conditional complexions), so in a sense Klein unified Galois theory and information theory (an achievement obscured by the historical oddity than on Earth, information theory appeared -later- than Kleinian geometry!).

Next, we can consider finite analogues of Euclidean geometry, in which new concepts "emerge", for example the euclidean notion of "distance". (Here, concept is also being used as a technical term; see Wille's theory of concepts in the undergraduate textbook by Davies and Priestly, Introduction to Lattices and Order. See also "closure operator" and note that "fixset of" is a closure operator.) Generalizing in a different direction, we obtain geometries in which another new concept make sense, "area". Or going off in yet another direction, we can obtain symplectic geometry with all the "emergent concepts" characteristic of that notion of geometry.

Before anyone asks, yes, these ideas should provide a useful tool in investigating Wheeler's old suggestion that geometry itself should emerge from more primitive notions, even from logic. I mentioned Fraisse theory, a topic in model theory in which one studies how properties emerge in first order logic from (essentially) Klein's point of view. These are very general ideas which apply to any group action, in particular any smooth action by a Lie group such as any of the "classical groups" studied by Weyl, but they are interesting only for nonabelian groups. Someone ought to write a book explaining all this good stuff.