Why is the Oort cloud spherical? Having read up a bit about the Oort cloud hypothesis, I find I don't quite understand this. Why is the Oort cloud spherical while the rest of the solar system is disk-shaped?
 A: I actually answered a related question a couple of days ago on Astronomy. Small world!
One of the important properties of the Oort Cloud is that objects in it are not strongly influenced by the Sun. After all, its inner edge is roughly 2,000 AU away - 300 billion kilometers from the Sun. The Sun's gravitational influence in that region is rather weak, so the Cloud can be easily subjected to other effects.
Galactic Tides
The first big effect (which stackErr mentioned) is the galactic tide. Tidal forces can stretch an object into different shapes. As Wikipedia puts it,

Galactic tides may then deform an otherwise spherical Oort cloud, stretching the cloud in the direction of the galactic centre and compressing it along the other two axes, just as the Earth distends in response to the gravity of the Moon.

So any disk-shaped object this size this far away from the Sun would inevitably become stretched out into an oblong or spherical shape.
Nearby Stars
The outer limits of the Oort Cloud may be over 1 light-year away. By comparison, the distance to the nearest star system (Alpha Centauri A/B and Proxima Centauri) is roughly 4 light-years. Once again, the weakness of the Sun's gravity means it is easier for the Cloud to be deformed. Passing stars long ago could have contributed to its shape.
As this picture shows,

Stars can and will come closer than that - just within the next 80 millennia! Within the past 4 billion years, odds are that we've had a lot more closer encounters. So any stars within a light-year or so would have had a pretty big impact on the Cloud.
Also, some of the bodies in the Oort Cloud may have originated in other star systems, or from somewhere completely different. Interstellar comets and Exocomets could have been captured by the Sun. This would mean that they wouldn't have to have orbits in the plane of the rest of the solar system - they could travel in orbits tilted at odd angles.
