Radiation comes in several varieties, of which the most important is light. Light is not matter, and so is not radioactive. Also common are electrons (beta particles) and helium nuclei (alpha particles), and Plain vanilla protons. None of these are radioactive, but they can easily have enough energy to become so.

Space suits are white because it reflects visible light, making it
(1) cool, in the bright, hot, sunlight (which is mostly in the visible frequencies),and
(2) a white space-suit makes the astronaut easy to see because it reflects so much.

For the most part, I believe that cosmic radiation should not be radioactive. Cosmic radiation can be either photons(light particles) or cosmic rays, which are atomic nuclei.

Most cosmic rays are protons (hydrogen nuclei) or alpha particles(helium nuclei) but there are also a small amount of heavier particle cosmic rays. A small fraction of the heavier particles could theoretically be unstable or radioactive but chances are that any radioactive particle would collide with something or otherwise decay into other particles before anyone would notice.

Since there is no air in space to carry heat away from objects, things can heat up very quickly if the sun is shining on them. Now it turns out that the sun shines mainly with visible radiation (which is why we can see it)and things that are at 300 K (27 degrees C) radiate in the infrared (not visible). The beauty of some white paints is that they radiate much better in the infrared then they absorb in visible light. So when the sun shines on something that is white, it absorbs a small fraction of the incoming radiation and then re-radiates it all away if the object is at a temperature around 300 K, which is the perfect temperature for someone who is used to being on Earth.

Can you think of why the visors on space suit helmets are coated with thin layers of gold or other metal rather than being clear?

Cosmic radiation usually refers to many different types of radiation. Usually the most common is the cosmic microwaves which are not radioactive. However there are probably some gamma rays (radioactive waves) that do exist within the realms of cosmic radiation.

I believe the color of space suit kind of matters which is why it is white. An astronaut wants a suit to reflect more light than it absorbs. Think about a choice of T-shirts you can wear on a hot summer day. If you wear a black T-shirt the shirt will absorb a lot of heat and you may feel warmer than if you wore a white T-shirt that day. For more information you can check out these websites:

Physlink.
Spacesuit.