Does a black body have a higher temperature when moving away from us at high speed? When a black body moves away from us at high speed the average velocity of its constituents will be higher. This means the body's temperature has increased. The whole spectrum of the wavelengths though will have been increased when arriving on earth, suggesting the body's temperature has décreased. Now my question is which of the two is the case?
 A: No, the temperature of the body would not be higher. 
The speed of the whole body doesn't affect it's temperature. Only the internal/chaotic movement of bodies atoms affect the temperature.
The mechanism of emission of electromagnetic waves by hot body is roughly like this: body's atoms oscillate (or move and collide with each other in gases), so they move with acceleration, and because the charged body moves with acceleration it emits electromagnetic waves. Let's take a body at 0 temperature and let it move as a whole with some speed. Even though each of it's atom has kinetic energy all of them move with the same speed, they wouldn't collide and wouldn't emit any radiation.
Now if this body collides with some obstacle and stops, total kinetic energy of it's atoms will remain the same (well actually it would probably decrease because some energy would be transferred to the obstacle), but now the atoms would move chaotically and the body's temperature would increase. If the meteor hits Moon's surface there is a pool of red hot melted stone!
A: lesnik's nice answer explains why the speed of the body does not affect its temperature.
The second part of your question has to do with the perception of a stationary observer, who, due to the Doppler's effect, will see all the wavelengths increased. 
Should the observer conclude that the temperature of the body moving away is lower than its actual temperature or will the observer be able to tell that the wavelength increase is due to the body moving away? 
I think the observer should be able to determine the correct temperature of the body and conclude that the body is moving away by analyzing the spectrum of the received radiation. This would be possible because the shape, not just an absolute frequency of the black body radiation, is unique for each temperature. 
Since the transformation of the spectrum curve due to the Dopper's effect is linear $(f=\frac c {c+v} f_0)$, the unique shape of the curve and the temperature information it carries will be preserved.     
A: If "high speed" means at some fraction of the speed of light, then the blackbody will appear to be cooler.
First off, as lesnik notes, the idea of measuring the temperature of a BB is basically a comparison of the average KE compared to itself. What you get is a curve, and that curve's shape will be independent of the overall gross velocity of the object. So in that respect, the measured temperature would not change.
However, at some point redshift/blueshift comes into play. In your case you have set it up so it's moving away, so it's redshift. So instead of seeing something that is, say, red-hot, the redshift will cause it to appear infrared-hot, which is, by definition, cooler.
In fact, it is precisely this effect that we use to measure the velocity of astronomical objects. In practice, stars are not perfect blackbodies (but pretty close) and we can often pick out specific spectrum lines. We assume (and have lots of evidence to conclude) that those spectral lines have the same frequency in distant stars as they do here on Earth. Thus, by measuring this shift, you can calculate the radial velocity.
