Are you able to calculate whether, by any chance, it will come close to any of the NASA probes around Jupiter, Mars, Venus, etc...? What is its closest approach to the JWST?

The closest it will come is Mars, but when I say close these are quite literally astronomical distances, about 0.2 au from Mars. This is about 75x further than the moon is from the Earth.

If it is an inactive rock, then we will not see it as any more than a point of light during its visit.