I'm going to answer these in no particular order.
Would the station have to orbit the point or could it just occupy the exact point?
L4 and L5 are stable Lagrange points. If an object at one of them was perturbed by some other body such that it moved away slightly from the point, it would move back to the point, thus ensuring stability. A good analogy is to picture these points as bowls, places of stable equilibrium. Imagine that the space station is a ball in the bowl. If the ball is somewhat jostled, it may roll up one of the sides a little, but it will come back down to its original stable position. So it's possible for an object to occupy the exact (well, it does depend on what you mean by "exact") point.
This is not true for all perturbations. There is a limit to the stability, generally reached if another body orbiting the Sun is massive enough and close enough. However, this is not the case for the Earth-Sun L4 and L5 points.
Orbiting space dust.
It has been conjectured that there is dust at these Lagrange points, called Kordylewski clouds. They are quite insubstantial and not at all dense; in fact, they are extremely hard to observe from Earth and their existence has not been definitely proven. Indeed, evidence for them is quite slim. The clouds might also be in an unstable orbit, again because of outside perturbations.
You can find a mention of the clouds here, though they are not referred to by name.
L4 has an asteroid, 2010 TK7, orbiting it. Would this be an advantage or a disadvantage?
2010 TK7 has quite the odd orbit, moving closer to and farther from Earth over time. It oscillates in a way that takes it far enough away from L4 for it to be practical.
The other consideration is that 2010 TK7 has a very high orbital inclination, as illustrated here:
Image in the public domain.
This makes it very hard to access, so it would not be a good resource for a space station at L4
Heavy shielding to protect electronics and crew from solar radiation.
As far as I know, the precautions taken here would be the same as would be taken for any orbiting space station. The only differentiation factor between a space station at L4 or L5 and a space station orbiting Earth is that the space station at the Lagrangian points would always have one part facing the Sun; it would never be in shadow. This shouldn't pose a huge problem, though; simply rotate the space station.
Heating / Cooling.
See the above.