Good question! I had forgotten about Jupiter’s Trojans and Greek asteroids!

I went and checked the definiton of Clearing the Neighborhood by IAU, emphasis mine:

The phrase refers to an orbiting body (a planet or protoplanet) “sweeping out” its orbital region over time, by gravitationally interacting with smaller bodies nearby. Over many orbital cycles, a large body will tend to cause small bodies either to accrete with it, or to be disturbed to another orbit, or to be captured either as a satellite or into a resonant orbit. As a consequence it does not then share its orbital region with other bodies of significant size, except for its own satellites, or other bodies governed by its own gravitational influence. This latter restriction excludes objects whose orbits may cross but that will never collide with each other due to orbital resonance, such as Jupiter and its trojans, Earth and 3753 Cruithne, or Neptune and the plutinos.[3] As to the extent of orbit clearing required, Jean-Luc Margot emphasises “a planet can never completely clear its orbital zone, because gravitational and radiative forces continually perturb the orbits of asteroids and comets into planet-crossing orbits” and states that the IAU did not intend the impossible standard of impeccable orbit clearing.[2]

Trojans and Greeks orbit Jupiter’s LaGrange points in a stable orbit and so they are governed by Jupiter’s gravity. You could say they’re really weird moons orbiting semi-stable points Jupiter creates.

No, because if it’s a proper planet it will clear its orbit.