Do both space and a moving body get length contracted in special relativity? I suppose I'll start off with an example of what confuses me. 
When considering how magnetism arises from special relativity, one example is often used. Consider a wire at rest with net charge density 0, and an electron outside of the wire. Now run a current through the wire such that the electrons inside move at speed $v$, and move the electron outside at speed $v$ parallel to the wire. The electron outside will experience a magnetic force and begin to move. From the reference frame of the electron, however, the electrons within the wire are stationary and the protons are moving with velocity $-v$. Thus, the protons get contracted, and so the charge density of the wire increases, and so the electron outside experiences a force toward the wire.
I get confused at the "the protons get contracted" step. It seems like what is being described is that charge density increases because the space between the protons is getting contracted, not the protons themselves. But I've heard that moving bodies themselves get contracted due to length contraction.
Another example is that of a spaceship traveling towards a distant star at .5c. I have heard that from the frame of reference of the spaceship, the distance between the spaceship and and the star gets contracted. But from the spaceship's frame of reference, all it observes is a star moving towards it, and so should see the star contracted. Why is the space between the spaceship and the star getting contracted? The observer can't tell that space is moving, after all, right?
I guess more generally I'm confused about how length contraction applies in different reference frames, and on what it applies to.
 A: Not being able to comment, please take this as one, not as  an answer.
I think you are forgetting the point that there is NO absolute space or time, so  thinking  protons get contracted is wrong in another way, because to you, everything looks normal. 
So if there are 10 different people, each in their own reference frame due to different velocities, everyone one of them thinks he/she is perfectly normal, and it's the other nine that have contraction "issues".  
In your title question, you ask do both space and length get contracted? You are worrying about nothing really  here. Think about how we measure space, we do it with a ruler. So if the ruler contracts, space is judged to contract, again not to you, but to the other guys.
So far, so good,  assuming that you are normal  and everyone else is lenght contraction and time dilated, but in SR, you need to reconcile your measurements with everyone else. 
This is where the Lorentz Transformation balances the books.
A: Length contraction is simply the the spatial effect of the Lorentz transformation, the fundamental transformation between the space and time co-ordinates measured by two relatively moving, inertial observers in flat spacetime. It applies to any length, whether the space concerned be occupied by protons, elephants, boiled eggs or no matter at all. 
If you are one of the observers looking at a laboratory of the other, then suppose you see a spatial interval in that laboratory that the other observer measures to be 1 unit long, then comparable measurements by you on that moving interval will deem it to be $1/\gamma$ units, i.e. $1/\gamma$  times as long as you would measure it were there no relative motion. The same holds for the other observer; he/she will measure a unit length in  your frame to be $1/gamma$ as well. For some measurement methods (e.g. measurement of angular subtense with a theodolite and inference of length from this value), you will need to account correctly for the propagation delays in the light needed to effect the measurement - this consideration leads to the peculiar and highly interesting result that a boosted sphere still looks spherical but is in fact ellipsoidal once the light propagation delays are accounted for, as discussed in the answers to the Physics SE question, "What does a sphere moving close to the speed of light look like?"! But, naturally, the same contraction is still found to hold once delays are correctly considered.
A: 1 wire: The protons must contract, the empty spaces must contract, whatever is in the wire must contract, so that the wire can contract.
2 space: Same as 1, but wire is replaced with something appropriate, like galaxy.
Or maybe we imagine a measuring stick between the spaceship and the star. If one end of a measuring stick is attached to the star, then the people on the spaceship say that the stick is moving and it is contracted. If one end of a measuring stick is attached to the spaceship, then the people on the spaceship say that the stick is not moving and it is not contracted. 
A: WE EXIST WITHIN ONE UNIVERSE.
However, let's say we have 10 identical spaceships, and that they each are moving within this universe at different velocities. Now imagine that we also have two objects floating in space at a maintained spatial distance apart. The observer within each spaceship will measure the depth of space that exists between these two objects, differently.
Each observer obtains a different result when each measures this same spatial depth.  This does not mean that 10 different universes exist. If you accelerate to a higher velocity, and thus now measure a different spatial distance between the two objects, this does not mean that you have changed the spatial properties of the universe.
Your small spaceships acceleration does not somehow alter the spatial width of the entire universe.
Thus it is to be understood that it is the measuring of spatial depths that changes, not a changing of the actual spatial depths themselves.
