Is temperature in vacuum zero? From Wikipedia entry on Kinetic Theory

The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. 

Now if I remove all the particles from the box shown below will the temperature be zero?

 A: The notion of temperature doesn't make any sense in complete vacuum (meaning the absence of all objects). It only makes sense as a description of how much some objects wiggle around.
To discuss the thermometer problem, one first needs to know what it means to measure temperature. "You just insert thermometer wait a little and you're done", I can hear people say. Well, not quite. What happens microscopically?
In the simplest case, if you want to measure temperature of a sample you need to attach thermometer to it. Molecules of these objects will interact and eventually will come into thermal equilibrium. Thermometer then has some calibration that tells you that so-and-so temperature corresponds to so-and-so much wiggling of its molecules. Well, it should be obvious that for this to work, concept of thermal equilibrium is essential. But you won't get thermal equilibrium if there are very few molecules of the sample. In particular zero.
Also note that the contact of surfaces is not the only way to attain thermal equilibrium. Any heat transfer process will do and that means any interaction. So you can try to measure the temperature e.g. by electromagnetic radiation. Well, if you insert such a thermometer into completely empty box then depending on the box's properties the electromagnetic radiation will either leave completely and thermometer will show zero or the box would trap radiation and the box would no longer be empty (it would contain photons). In any case, what you are measuring now is not temperature of the vacuum but rather the EM transmission properties of the box.
To summarize: problem of measurement is not a trivial one and it has actually lead physicists to great many discoveries. Noting that you can't simultaneously measure position and momentum gave rise to quantum mechanics. Noting that couplings of our elementary particle theories depend on the energy you input into measurement gave rise to renormalization and better understanding of quantum field theories as a whole. So it's always important to think about what are you actually measuring microscopically.

Now, let me talk about some related things for a bit.
Consider that box full of gas again. As you'd lower temperature of the walls the molecules would transfer their energy to the walls and become slower. Now, you can imagine that by doing this for a long time you'll eventually reach zero temperature and all the movement will stop.
In reality, this is not possible because you'd need infinite time to reach that temperature. And even if you had that time, you have to take uncertainty principle (you can't know the position of an object absolutely precisely) into account. Actually, cooling is a big field of physics in itself and entails various extremely sophisticated techniques that are very close to 0K.
Also note that in reality there is no such thing as vacuum (again in the above sense) because of quantum fluctuations.
A: According to the above definiton the answer is not defined mathematically, the average kinetic energy is  
$\displaystyle 1/N \sum_{i=1}^N m_i*v_i^2/2$  
If we plug in N=0 we see that it blows up  
Now if we had some other definiton or the gas was not quantized maybe we could take the limit as N->0
Edit:
This answers your question if you define a thermometer as something that measures temperature, and you have defined temperature as above.
A: You cannot remove all the particles from the box. Even if you remove all atoms, still there vill be photons inside, which carry their own kinetic energy. The photons are generated by the box walls and reach with time thermostatic equilibrium with the walls, so the temperature inside the box will be the same as the temperature of the box walls.
Any body placed in such box will be gradually reaching the same themperature as that of the box, through radiation, even if there are no atoms inside.
Only if the box walls are at absolute zero there will be no photons inside.
A: Even if you remove the particles, there will be a thermal radiation coming from the borders. This way a thermometer placed inside will eventually show the temperature of borders.
A: There's no temperature.
If we use the following definition "temperature is the average
kinetic energy of the particles". Then no particles - no temperature.
As the first sight this answer doesn't seem to be good enough, but if you want
to calculate "average spin" or "average charge" those parameters will have no sense
if there's no particles to calculate data on.
A: Vacuum does not contain radiation nor other electromagnetic fields.
Stationary and constant velocity vacuums do not have any temperature. Accelerated vacuum has a temperature - the Unruh temperature:
$${\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}}}$$
It is very weak. In the acceleration of Earth's gravity it is only $4×10^{−20}$ K.
