It's not a stupid question. Space is already highly radioactive, with lots of protons and bigger nuclei from outside the solar system flying around in all directions, along with a ton of proton radiation from the Sun.
Most nuclear spaceship designs have a long stick with the reactor at one end and the crew on the other, and some kind of shielding in between. Since you only need to shield a cone, it can be quite effective.
Yes, but none of that changes the fact that when we talk about fission rockets, we have to think about nuclear safety on top of everything else. (All of your comment is thinking about nuclear safety.)
And if we use them inside of a planet's magnetosphere, we have to think about environmental contamination too.
Yes, but space is already full of radioactivity. An incredibly large amount of it. The average range (outside of the upper Earth's atmosphere) would be equivalent to getting somewhere around 10 chest x-rays per day, every day.
More or less, yes. Space is so large and so rarified and so riddled with energetic particles and photons that the contribution of a few measly rockets is of no concern.