Privět. These are real-world questions that NASA, Russian/Soviet space program, and others of course had to be solving – if you kindly believe that astronauts are real – when they were designing space suits, see e.g.
Actually these 8 pages about space suits could be more useful (buttons 1-8 are at the bottom):
Space suits only give the astronauts oxygen – from the spaceship – and have to remove carbon dioxide that the astronaut breathes out. Neoprene and other layers of fabric isolate the astronaut at the inner side. The outer side is "white" – it is made out of a highly reflective material so it doesn't really absorb much of the intense solar radiation that you were sensible worried about.
There's extra heat from sweating. Gemini and Mercury programs used cool air. Since the Apollo program, NASA has been using water cooling. All the required material has to be available in the space suit and/or the spaceship. You essentially propose to equip space suits with an active fridge. In principle, it's a good idea but it's hard to quickly get rid of the heat without a "reservoir", anyway.
A human consumes 2,000 kcal a day – the heat needed to warm 2,000 kg of water by 1 °C (or 200 liters of water by 10 °C, and so on). That's the usual "recommended nutrition value". In normal units, that's about 8,000 kJ a day. Most of this energy ultimately ends up as heat. When divided to 86,400 seconds, you get about 90 W. Well, at rest, a human actually produces about 70 W (70 Joules per second) of heat. It's like a classical light bulb.
However, you don't have to remove all this heat manually. Much of it is just radiated away by thermal radiation.
Of course, spaceships have to deal with much greater amounts of energy to move, cool the engines that heat up, and so on. Fuel for a space shuttle includes a ton of liquid hydrogen plus tens of tons of liquid oxygen – that's energy comparable to 140 GJ or so when burned. If it were used to replace the heat from an astronaut, it's enough for billions of seconds.
Space shuttles were cooling inner surfaces of nozzles by liquid hydrogen. Note that the latent heat of hydrogen is 461 kJ/kg. I said that a human produces 8,000 kJ of heat a day – it is just the vaporization of 20 kg of hydrogen (per day) if you needed to manually remove all the heat which you don't have to.
The International Space Station where astronauts spend years needs a long-term solution, a cooling system that was developed by Boeing and contains many components. In some of them, ammonia is used.
There are lots of engineering issues but the things work when the dust is settled and bugs are fixed. You should understand that while the astronauts need oxygen and other things, they're not really "depending on every second of life support" when it comes to the temperature. When you're isolated enough, you may survive at the North Pole or the equator. The outer space (with a space suit) isn't too different in this respect.
See also a question on oxygen production at the ISS etc.:
How do they produce air on the ISS?