Is the centre of axial precession perfectly perpendicular to the ecliptic? It is my understanding that the Earth's axial precession is such that the direction of the North pole pointing to the celestial sphere traces out a circle with a period of roughly 26,000 years. My first question on reading this was, where is the centre of that circle? In other words, what axis is the Earth precessing about? I had assumed this would be somewhat arbitrary but have since learned that it points towards the north ecliptic pole. Am I then correct to assume that planets naturally tend to rotate in the same plane as their orbits, but the effects of precession caused by other massive bodies cause the axis of rotation to oscillate around the north ecliptic instead?

If so, then what is going on with Uranus and it's unusually high axial tilt? Is this another phenomenon or does Uranus' rotation axis also precess about it's own ecliptic north pole? Albeit with a much larger amplitude.
The answer lies in the cause of the precession. Spinning objects don't simply precess of their own accord, it is the effect of other bodies acting on the rotation object to induce a torque.

In the case of the precession of the equinoxes, the precession is caused by the other bodies in the solar system, most significantly, the sun and the moon. Because the Earth is slightly oblique (the equatorial radius is greater than the polar radius), there is an equatorial bulge. Further, because of the tilt of the Earth's axis, this bulge is tilted with respect to the ecliptic. The gravitational pull of the sun acts more strongly on the side of the bulge closer to it and creates a small torque. Similar effects take place due to the presence of the moon, the combined effect is known as lunisolar precession. Other near by massive bodies also play a role.

A complete understanding of how/why this torque produces the effects of precession requires an understanding of angular momentum in general and is relatively trivial given that.

To answer the question explicitly, the centre of axial precession points roughly along the ecliptic pole because the effect is caused by the sun (which is always on the ecliptic) and the moon (which is very close to the ecliptic).