No asteroids or planets will be close
Actually, planets orbiting out of the plane of the star's rotation, or out of the plane in which the majority of the planets in the system orbit, is fairly commonplace in Star Trek. Virtually every star system schematic we have seen in Trek has featured at least one planet orbiting at an odd angle...
Whether asteroids would be in a nice plane depends on how old they are. In Trek, there are plenty of cataclysms that could create asteroid rubble late in the life cycle of a star system, mere geological moments before our heroes arrive; fully spherical asteroid "shells" around stars might be quite common.