How do we know the expansion of the universe is "of space" and not "in space" or "into space" or another less intuitive arrangement? How do we know the expansion of the universe is "of space" and not "in space" or "into space" or another less intuitive arrangement? For example, what implicit and explicit assumptions underlie the conception of expansion of space, itself? What extent are they consequences of what 'grade' of scientific knowledge? E.g. for Space to Expand, space needs to be a physical object? is there direct evidence of this? Any help would be much appreciated.
 A: The reason it's thought of as expansion of space rather than just things moving farther apart through space is that the math of general relativity describes it that way, and GR has been well-supported by experiments so far.
GR is all about curvature of spacetime, and curvature of anything can be determined by how we measure distances. A lot of the math in GR is analogous to the math that maps a globe to the Mercator projection. In cartography there are formulas for converting the length of a vector on the flat map to the length of the actual vector at that point on the globe. A one-mm vector near the equator will correspond to a longer vector on the globe than a one-mm vector near the top or bottom of the map. Knowing how this metric changes is enough to determine the curvature.
The vectors you look at in general relativity are 4-vectors that involve space and time. The equations of GR tell you how to calculate the spacetime "length" of those vectors, at any point in spacetime, just as the equations of the Mercator projection tell you how to calculate the on-the-globe length of a vector at any point on the paper map. And the metric is pretty much the same for all parts of space at a given time (on a large enough scale to average out galaxies), but it changes from the past to the future in a way that means things are farther apart, on average, in the future. Since a change in the metric represents a deformation in space and time, we say space is expanding.
It all comes out of exactly the same math that says that (for instance) time slows down near a black hole. Originally I said that the Hubble expansion would cause objects initially at rest with respect to one another to drift apart, but apparently that is true, but only because of dark energy (i.e., with the cosmological constant). See, for instance, this paper. It can be tricky to get this stuff right.
If you want to see some actual math, check out the FLRW Wikipedia Article, and the other articles it links to. 
A: Expansion of space is allowed by General Relativity. Which has made quite specific predictions that passed. So it seems valid to have solutions with an expanding space on the table (and leave it to observation to exclude them or to favor them).
And there are two different situations where it comes up. 
A cosmological context where the expansion is the whole universe getting bigger and the universe is effectively expanding into the future (because that is what universes do) and the future happens to be bigger than the present.
The other context is the most benign case of a star or planet forming. All curvature in general. When a star forms for example there is one type of curvature outside and s different inside. And it is natural for spacetime to curve in the outside type all on its own and matter allows different types of curvatures to join together.
As the star collapses the region where the two types of curvature join together moves. And thus the outside type extends itself deeper into where the star used to be, creating new and stronger curvature. However this process of star formation actually created space.
You can imagine people at the center staying at the center measuring how far away the surface is and they see it decreasing. And imagine people at the outside staying where the surface starts. And they measure the distance too and see it increase.
It increases more for the people outside than the people inside. A good analogy is to imagine a funnel as the outside type curvature and a flat disc as the inside type curvature.  If you decrease the radius of the disk by 1% then you can place it farther down in the funnel but the funnel distance has to travel down the funnel as well as in towards the smaller disk.
That mismatch is newly created space. But entirely local, not cosmological. But it is an everyday everywhere phenomena, new space forming as every star forms. 
