Is the kind of physics proposed for "warp drive" related to the way that space really is expanding? As I understand, some parts of Universe really are moving faster than light -- is this expansion something we think we can create artificially?
And if we can do this artificially, could it be done a on very small scale in the near future? Or is it possible that space is already being affected in this way in, say, particle accelerators or during nuclear explosions?
 A: 
Is the kind of physics proposed for "warp drive" related to the way that space really is expanding?

Related, yes, in that both are based on Generated Relativity, but the two are fundamentally different (specifically, they have different units).

As I understand, some parts of [the] Universe really are moving faster than light

well, kind of. Depends on what you mean by "moving": the distance from us to certain places in the universe (as measured now), is increasing faster than the speed of light, yes. But thinking of this as the "speed of the universe's expansion" would be misguided, since this variation of distance cannot be meaningfully interpreted as the "velocity of something" within GR.
But, our model for the expansion of space (ΛCDM cosmology) describes the expansion as being spatially uniform on large enough scales.
The typical analogy is to think of dots drawn on an inflating balloon. Each of these dots is stationary, it's the distances among the dots that are increasing.
Also, no thing is moving through space at a speed larger than $c$: cosmological expansion is locally compatible with Special Relativity.
So, the meaningful quantity to describe the expansion is not a speed but an expansion rate: something in the form $(\Delta D / D) / \Delta t$, where $D$ is some (large, cosmological) distance, and $\Delta D$ is is variation across a time difference $\Delta t$.
Notice that the units of this are [1/time].
This is a somewhat loose description of the Hubble parameter.

is this expansion something we think we can create artificially?

Maybe, in the far future! Surely not with current tools.

And if we can do this artificially, could it be done a on very small scale in the near future?

I'd venture that if we did recreate it artificially it would likely be on small scales compared to the cosmological ones we currently observe expansion at! But this is quite speculative, of course.

Or is it possible that space is already being affected in this way in, say, particle accelerators or during nuclear explosions?

So, if "this way" means FTL then no, with quite good confidence, although people are still looking.
If "this way" means to have some general-relativistic effect in particle physics experiments, then still no unfortunately, and we might not see anything of that sort unless we build detectors with much higher energies.
Regarding the warp drive: it is a theoretical construction within GR with the following properties

*

*it entails no local violation of the speed of light,

*it allows one to move arbitrarily fast from A to B, both according to the one moving and to people outside (the latter is the difficult part!),

*it requires negative energy.

The first thing sounds like what I was writing before, but that's just because it's a GR requirement, and both models are GR-compatible.
The second thing is not happening in cosmology!
The third thing is what makes this "impossible" (i.e. we don't know of any current materials or techniques to make negative energy in any substantial amount, and you'd need a black hole's worth).
A: At the current moment, we don't even know what dark energy is which is the reason the universe expands faster than the speed of light. There is not really a way we could recreate it per se but there are talks about how we can create an effect similar to dark energy which has some kind of "negative energy effect".
There has been talks about how Harold White with his Eagle Works Laboratory has been working on like a potential precursor to the warp drive and might use the Casmir effect to cause some kind of dark energy like effect which could maybe used on warp drives. However, there has not really been any real success and for the moment it is pretty much impossible to harness such a faster than light expansion.
In the end, such a thing has a really low chance of existing but there might be a tiny chance that something might be discovered. Even if it is discovered, harnessing it is very difficult thing to do.
There is not really any other known way to create a faster than speed of light flow of space(maybe aside from black holes).
