Conservation laws in the finite element method for domains with curved boundaries

There are at least three different disciplines involved with the question: Let's start with the last one. Stationary heat conduction inside a solid body is described by a (set of coupled) partial differential equation(s). However, it is clear from the outset that the infinitesimal quantities in such equations must be considered as a mere illusion, an idealization to be precise. Especially the differential volumes are not really infinitely small. Far from that! They should contain a lot of molecules, if we want the approximations, yes, the approximations of Classical Analysis to be considered as valid. There is a beautiful text about this (thanks to Jean Baptiste Perrin) in Benoit Mandelbrot's 'The Fractal Geometry of Nature', first few pages (6,7). Unfortunately, I don't have the book in my shelves and I can't exactly quote. Anyway, while thinking about it, two things should become clear, if PR is to be adopted as our truth frame of reference:

  1. The infinitesimal volumes of Classical Analysis are too small when compared to PR
  2. Likewise the finite elements, differences, volumes of Numerical Analysis are too large
With a pinch of salt, because boundaries between the disciplines are not always crisp. But I hope the message is clear: Classical Analysis is not "better" than Numerical Analysis per se. Neither is NA superior to CA (but it seems that the majority of people does not think that anyway).
If I understand it well, the OP's questions boil down a great deal to the following: trying to comprehend what the relationship is between numerically and analytically. I think the best approach is that CA and NA are quite different worlds, two entirely different ways of doing mathematics, though (hopefully) corresponding with the same physical reality in the end.
Perhaps the most important difference between the analytical world and the numerical world is that their function spaces are completely different. In the analytical world, functions most of the time have continuous n-th order (partial) derivatives. In the numerical world, functions are often piecewise linear. This means that continuity already stops at low orders of the derivatives. With all this in mind, it's probably less difficult to accept that equations 4) 5) 6) belong to CA and dfferent equations 4') 5') 6') belong to NA.

Question 1.

But let's end the prose for a while and calculate. As a generalization of the OP's tetrahedron, consider one with vertex $(0,0,0)$ at the origin and three vertices at the surface of the unit sphere; let the latter form an equilateral triangle with edge lengths $=a$. Then with some elementary geometry and algebra we find for the distance $\,h\,$ between the origin and the midpoint of any such equilateral triangle: $$ h = \sqrt{1-a^2/3} $$ For the sample tetrahedron $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$ we calculate $\,h=1/\sqrt{3}\,$.
Now there are two ways to look at the OP's dilemma concerning $\,g(x,y,z) = \sqrt{x^2+y^2+z^2-0.9}\,$:

  1. Stay within the world of NA and accept that the triangles employed at the surface of the unit sphere form a piecewise linear interpolation of the values at the vertices and nothing else. Then the solution is that $g(\mbox{midpoint})$ is the mean ($1/3)$ of the $3$ function values $\sqrt{0.1}$ at the vertices, which is simply $\sqrt{0.1}$.
  2. Staying within the world of CA and demanding that the argument of the square root remains positive. The common trick to accomplish this is mesh refinement, in our case as follows: $$ h^2-0.9 \ge 0 \quad \Longrightarrow \quad 1-a^2/3-0.9 \ge 0 \quad \Longrightarrow \quad a\le\sqrt{0.3} $$

Question 2.

Concerning the second question, it is indeed the case, or rather: it should be the case that conservation laws hold in the analytical world as well as in the numerical world. There is one major difference, though. The conservation laws in the analytical world hold for an infinitesimal volume as well, but the conservation laws in the numerical world only hold for finite volumes. I do not consider these isues as trivial and have worked out some in answers to the following question:

Perhaps there are other ways, but I have done it as follows. Conservation laws hold by definition if finite volume methods (FVM) are employed. So if we are able to to establish a mapping between finite elements and finite volumes, then it is for sure that conservation laws also hold with FEM.