I will divide this answer into 3 parts
Jupiter's radiation isn't actually a risk to us. Jupiter's radiation is simply a result of Io's charged volcanic plumes forming a radiation belt around it. Our Earth does possess these radiation belts (though they are generated by solar wind and not volcanoes). We call them as Van Allen Radiation Belts.
Jupiter's radiation belts are only dangerous to astronauts on a moon with no atmosphere, i.e. Europa or Ganymede. The lack of an atmosphere makes it susceptible to Chernobyl-level radiation.
For the Earth, this is not so, because we have an atmosphere, which is pretty much opaque to radiation. However the radiation is a minor problem, compared to the most difficult problem we will ever face.
The Jupiter-Mars-Earth-Moon system will be dangerously unstable
Earth has a Moon, and Jupiter has more than 60 moons in its system. Our system would be unique because, Earth would be the only moon, which also has its own moon, i.e. a submoon system.
However, this means that the system would also be really unstable, as our Moon is still more massive than Europa. This would either mean that Moon would drag it into an orbit around the Earth (highly unlikely), or else it would redirect it to crash on Mars. Remember that although most of Jupiter's moons are massive than our Moon, our Moon is still comparatively massive *(for e.g. Ganymede is only 2x as massive as our Moon).
Furthermore, the combined gravitational pull of Jupiter, Earth and Mars, proves to be too much for our Moon, and can trigger volcanoes to erupt on our Moon. This would imply that our Earth's radiation belt would become stronger and be a major problem for future space explorers.
Furthermore, Earth's magnetic field and Jupiter's magnetic field interact producing complex phenomena. Although Jupiter's magnetic field is already big, we must not forget that Earth too also has a large magnetic field, while its magnetosphere might stretch out to only 36,000-60,000 (6-10 Earth Radii), its magnetotail stretches out to 6 million miles, about 6 times the Sun's size. Pictures of Earth's and Jupiter's magnetic field-
As seen above, both the magnetic fields are already comparatively large. When you get multiple magnetic fields to interact with each other, you can get really complex phenomena, like magnetic eddies, and as Io's volcanic plumes enter Earth's domain they would generate strong auroras which would be intensely radioactive to flight passengers.
Also, as Jupiter, Moon and Earth exert gravitational pull on Mars, this can also result in Mars's core remelting and generating a magnetic field. Also, the Moon itself is now being intense flexed by Jupiter, Earth and Mars, generating a lunar magnetic field. The Earth is not intact, as Jupiter, Mars and Moon also pull on it, perhaps causing a stronger magnetic field.
When Jupiter's magnetic field, Earth's magnetic field, Mars's magnetic field and Moon's magnetic field interact, they no longer behave like a protective shield and instead behave like a giant, oversized particle accelerator, occasionally bursting in energy due to the combined effect of both the solar wind, Io's and Moon's volcanic plumes erupting trillions of tons of matter out into space. This forms an region of intense radiation which is potentially lethal for future space explorers. However I think I may be digressing a bit so let's move to the next point.
Jupiter's hill sphere is about 57 million km across as pointed out by @Resonating in the comments, but that doesn't mean that you would have a comfy space to place a orbit in. Gravity decreases gradually as you come closer to the Hills Sphere, and at the Langrange point, which is always at the edge of the Hills Sphere, the orbit is extremely unstable, like a pencil on its back-end, it can fall given the slightest disturbance. So we need to be much closer to Jupiter to get a (still unstable) proper orbit, same for Mars.
This question is significantly more difficult to answer, and it can vary. Jupiter's gravity at that distance cannot affect small objects like spacecraft, so I guess that my answer would be yes, though that would be backhanded by the fact that the intense radiation produced would make space travel extremely risky.