Is the metric expansion of space relatively uniform on different length scales? Is the Metric expansion of space relatively  uniform in space? In other words, loosely speaking, does expansion happens everywhere, and over a wide range of length scales. 
For example, the Hubble constant (say 70 km/sec per megaparsec) would be about 2.3E-05 m/s at 10 billion km. Neglecting numerous profound experimental difficulties, if it were possible to make some kind of measurement with a controlled experiment, over such a short distance, would we expect to see expansion locally consistent with the cosmological rate?
Assume the experiment is in a relatively empty area in space, where one is not distracted by large scale structure so that one tries to put all the expansion between those structures and not within those structures.
note: the question is about the expansion rate itself, not about how difficult it would be to measure. The question is also not about how expansion has been historically inferred from earth-bound observations of complex structures like galaxies. It's about the space.
 A: Such measurements have been done, using lasers reflecting off mirrors on the moon. See e.g the paper Progress in Lunar Laser Ranging Tests of Relativistic Gravity (Williams et. al. 2008) which established an effective limit on the expansion at AU scales that is about 80 times smaller than what would be expected if cosmological expansion applied within our solar system.
As John Rennie explained in an answer to this question, the expansion is a property of the FLRW metric, but the local distribution of matter doesn't match the assumptions for that metric (which hold well enough on cosmological scales). That doesn't prove by itself that a metric that describes our solar system doesn't have expansion, but the experimental evidence is that if it does it is much smaller than you'd expect from a simple extrapolation of Hubble's law down to AU scales.
A: 
Is the Metric expansion of space relatively uniform in space? 

No. 

In other words, loosely speaking, does expansion happens everywhere, and over a wide range of length scales?

No. 

For example, the Hubble constant (say 70 km/sec per megaparsec) would be about 2.3E-05 m/s at 10 billion km. Neglecting numerous profound experimental difficulties, if it were possible to make some kind of measurement with a controlled experiment, over such a short distance, would we expect to see expansion locally consistent with the cosmological rate?

No. 
Think of the raisin-cake analogy. The cake expands, but the raisins don't. And the moot point here is that space expands between the galaxies but not within. Because it's gravitationally bound. In similar vein space where an electron is, is electromagnetically bound, so that doesn't expand either. Ditto for matter in general. Another analogy people refer to is the balloon analogy. The balloon is getting bigger, and the skin of the balloon stands in for space. But note that one part of the balloon is quite literally "bound": the part where the knot is.  So the expansion isn't uniform. It's similar for space. Imagine you could grab hold of a piece of balloon skin and pull it out into a long tubular protusion then tie a knot in it. Then repeat such that the balloon skin is dotted with knots. Think of each as galaxy.   

