Does the earth's orbital plane change with time relative to the distant stars? Does the Sun follow the same path through the zodiac (ignoring proper motions of the zodiacal stars)?
The position of the ecliptic does change, due to perturbation of other planets, but only very slowly.
The invariant plane is the plane of angular momentum of the whole solar system. This is dominated by the orbital momentum of the planets, particularly Jupiter, and since other stars are much too far away to perturb the solar system, it is invariant with respect to distant stars.
The ecliptic does vary. It is currently decreasing by about 1.4 degrees per 10000 years, as indicated by this graph, (from Wikipedia). This variation is one hundredth of the variation in the Earth's rotational axis.