To add to this: Most of the heat the EVA suits deal with is generated by the human inside not the giant ball of nuclear fusion 8 light minutes away.
Solar radiation is roughly 1 kilowatt per square meter. Human beings generate about 0.1 kilowatts. A good suit will try to reject as much of that kilowatt as possible. Also your dark side will radiate heat but the temperature differential is much lower.
Suits are insulating for a reason. You want to prevent heating on the sun side and prevent too much cooling on the space side. Your body is essentially encapsulated in a giant thermos.
Cooling is achieved using a recirculating cold water system that is good for a few hours of body heat. Water is initially cooled by the primary life support system of the spacecraft before an EVA. Pretty much it starts off pretty cold and slowly over time comes up to your body heat. Recent designs use evaporative cooling to re-cool the water.
Life support systems are so cool.
Yes! The tiny number of particles are moving really fast, but there are very few of them. We are talking about vacuum that is less than 10^-17 torr. A thermos is about 10^-4 torr. The LHC only gets down to 10^-10 torr. At those pressures you can lower the temperature of a kilometer cube by 10 thousand kelvin by raising the temperature of a cubic centimeter of water by 1 kelvin. There is very little thermal mass in such a vacuum which is why temperature can swing to such wild levels.
This is also why spacecraft have to reject heat purely using radiation. Typically you heat up a panel with a lot of surface area using a heat pump and dump the energy into space as infrared. Some cooling paints on roofing do this at night which is kind of neat.