When we refer to the 3 K of temperature in space, we don't mean atomic vibrations. The so called temperature arises, when you look at the sky and measure
the radiation, which comes to us from every direction. If you cancel all stars, galaxies and other major light sources you will still "see" very isotropic microwave radiation. And this radiation is distributed like the radiation of a black body at a temperature of 2.73 K ($\approx$ 3 K).
So where does this radiation comes from? As you may know, we live in a accelerating universe, which means it is getting bigger with time. So if we reverse the time, we have a shrinking universe. But when every length scale
decreases, so does the wavelength of the photons in the universe. By that logic
the average frequency, and thus the energy, of a photon was large in the past.
Now there was a time, when the average energy of a photon was so high, that there couldn't be stable atoms, since every atom-photon interaction would be so
strong, that the electrons would instantly leave the atom. The universe had to
increase and cool until it reached a energy of about 3000 K. Now there could be
stable atoms. The photons we are seeing today, are photons which got scattered the last time at this very moment! But since many billion years have pasted, their wavelength has further increased and instead of photons with a spectrum of
a black body with 3000 K, we see a 1000 times cooler spectrum, which let's us
extrapolate the time of the last scattering to about 380000 years after the big bang (or 13.4 billion years ago).
Your second question can be answered under many different aspects. The first I personally find quite funny: actually you are not freezing to death, you are boiling! Since there is no outer pressure in space the boiling point of your body fluids would rapidly decrease under your body temperature. In the first moment your fluids will go partially into gas phase and will damage your tissue by the increasing volume and pressure. But don't worry, you won't explode.
In the next step your body fluid would have lost so much temperature and energy due to the boiling process, that they would actually starts freezing. But this
hasn't be the case for your hole body. As you have said, the only form of heat
exchange in space is through radiation, and our body looses only a negligible part of it's heat this way. But we absorb pretty much every radiation! So under direct sunlight your back would heat up to a few 100$^\circ$ C (above the boiling point of water) while your front is below 0$^\circ$ C (below the freezing point of water). Never the less, you would be dead in less then a minute, but you could die on both ways at the same time!