How do we know that gravity is a fundamental force, rather than an emergent one? 
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*How do we know that gravity is a fundamental force, rather than an emergent one?

*Also, what are the smallest masses and length scales for which we have measured it?
 A: The Particle Data Group's review Experimental Tests of Gravitational Theory, updated 2014, quotes separations as small as 0.054 mm [D. J. Kapner et al, 2007].  Those experiments measure the torsion of machined disks with holes in careful patterns on torsion pendula; the masses of the disks are quite small.
Here's an intriguing measurement of a second-order gravitational effect between modest-sized laboratory masses and clouds of atoms; it was just published Monday and I haven't read it yet. The smallest nonzero mass shown to have graviational effects is the hydrogen atom, rather than the neutron; bare electrons are charged, and positronium isn't stable. But those experiments have the Earth as the other mass; if you want the smallest product $m_1m_2$ it's probably one of the tabletop-gravity experiments above.
As for emergence: while we know of many forces that "emerge" from electromagnetism in condensed matter, we don't know whether any of the four "fundamental" forces are emergent in the same way or not.  You might enjoy Robert Laughlin's book A different universe which discusses the subject.
A: Rob has excellent answered part 2 of your question. Here's my thoughts about part 1.
We don't know that gravity isn't emergent! In fact there are several active research attempts to cast gravity as an emergent phenomenon. One of the hottest topics is entropic gravity which tries to derive general relativity from thermodynamics.
The basic idea is this. We know that spacetime is perceived as hot by some observers. For example black holes have a temperature inversely proportional to their mass. This suggests a link between gravity and thermodynamics.
It turns out that you can reproduce Einstein (and other) theories of gravity in a completely thermodynamic language. This is regarded as surprising or trivial depending on who you talk to! Some believe that this points to a thermodynamic interpretation for gravity at a fundamental level. Others think it's just a mathematical equivalence with little underlying physics at a basic level.
If you'd like to read more, I'd suggest this review paper.
Addendum: there's perhaps another reason to expect that gravity might be emergent. The AdS/CFT correspondence describes a bulk gravity theory which emerges from quantum field theory degrees of freedom on the boundary. Various people, principally Erik Verlinde have argued that gravity can be described as a statistical effect using this idea.
