Actually, they don't. There have been many nuclear space missions, and Cassini was the only one with a significant protest. And here's the kicker--the protest
had no effect. Cassini was launched anyway. The oh so incredibly powerful anti-nuke lobby has stopped a grand total of
zero nuclear space missions.
Face it, the anti-nuke crowd is nothing but a weak scapegoat here in the USA and more or less a non-entity in the Soviet Union/Russia. Ask yourself why the Soviets didn't use Orion. It certainly wasn't fear of anti-nuke protesters. No, it's a matter of ridiculous cost, technological challenge, technology risk, and so on.
Unless the "mission" is to blow up a spacecraft with a nuclear bomb, yes.
Here's one
link. Geoffrey Landis was kind enough to e-mail me a copy of the paper, but generally it's only available behind a paywall.
Braking at the target system is typically conceived of as done using Zubrin's interstellar medium brake, although I prefer faster (and more expensive) methods.
Currently, I'm developing a new concept I call "twin stream propulsion", which may be an improvement over my long time favored method of braking/return--relativistic kinetic impact powered rocket. Essentially, relativistic kinetic impacts can cause explosions with greater velocities greater than that of nuclear explosions (or even anti-matter). Using such relativistic explosions, it's possible to get around the limitations of nuclear propulsion.