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Fun fact: the temperature of space is actually thousands of degrees, but you would still freeze to death without protection.
(The actual answer is that atmospheric pressure is just as important as temperature in determining how "cold" something is)
The temperature of space actually is close to absolute zero, so quite cold. As the heat balance of an object there is mainly dominated by radiation, the object looses heat (~T⁴) but almost has no heat input from the surrounding if not directed to a star in sufficient proximity, e.g. the sun. The surface exposed to sunlight however, can become really hot.
It really depends on what you mean by temperature. You're both right, but both wrong depending on context.
Individual atoms and particles tend to have a lot of energy, but also there's almost no heat transfer into larger bodies because of the low density of those particles, so you lose more heat to radiation than you take in (unless you are in direct sunlight.)
Yes. One place in space has different temperatures. I would assume even individual particles are not distributed by a Maxwell distribution, so the concept of temperature is hard to apply. The background radiation has one temperature. If you add the sun, however, you already have a problem as the sun radiation is not in thermal equilibrium. So depending on how you look at it, you get different temperatures. The particles have a high energy, so also a high temperature. But they are so rare, that radiation is the dominant mode of heat transfer and determines the temperature of a thermometer placed in space.